00:00 - 09:0009:00 - 23:00

12:00 AM
Exactly what I would do

people had to put their own tacos together, of course. we weren't going to pay for that.

A hog roast is also a good addition to any event

oh, that would have been great.

My cousin had a hog roast at her wedding, that's all I cared about (except maybe for the free beer)
and my uncle's unprepared speeches
He was the top salesman for a large Swedish company every year for many years. His skill at successfully bullshitting his way through any presentation was incredible

that's great. we did not have anybody in our family with that talent, so we actually imposed a rule, no speeches. if you don't have a character who can really do one, they just bore people anyway.

12:06 AM
Good rule

we also had a thing, the venue would charge us more if we went over 4 hours. we were gone with 15 minutes to spare.
money dictates everything.

That's a shame

i didn't surpass my dad, who spent approximately $25 on his wedding. lmao that's great How'd he manage that Just signed the register with a few witnesses? no they had an actual priest! but he was married on an air force base. during the week, not a lot of people in the chapel. and no schedule of fees.$25 was for i think cake and something resembling champagne but not.

12:09 AM
Brilliant

it was during the cold war so the photos are pretty funny. the chapel had this panel of dr. strangelove style signs in the background that would light up if the russians attacked. stuff like "SAC ALERT" etc. helpfully, the russians did not disrupt his wedding.

Good guy Soviets
time to go bash my head fruitlessly against my probability notes

wear a helmet.

I'll need kevlar

12:42 AM
our wedding was in the brazilian rm in tilden. all the irish folks were sunburned after the brief exposure to sun.

yes, the notoriously harsh east bay hills sun. they would have perished at my wedding.
i turned a pretty good shade of red. i didn't want to ruin my suit by getting sunscreen on it.

12:56 AM
i was just happy that no international incident occurred. that was the next day.

my wedding was the first time my parents had seen each other since their divorce 15 years prior. i'm surprised it wasn't a nightmare.

Well, maybe you guys deserve guilty verdicts.

i only had one other math person at my wedding. low dimensional topologist.
i think 3 was as high as he would go. and only if he had to.

1:27 AM
Not much left with 1 and 2.

1:43 AM
if a cord gets tangled up i hand it to him and say "this is your job."

that's the plane and straight truth

Sounds more bent to me.

:-)

i got a response to one of my nasty friday letters today. they told me to pound sand. now i'm going to tell a judge on them.

i no longer have the energy for persistent fighting.

1:52 AM
I don’t relish conflict as much as leslie. More proof to @Andrew that we are very much not one.

well, generally i have been able to find a middle ground, but some folks just can't let go.

2:10 AM
if this wasn't my job i would hate this. i treat it like acting in a play.

i am not good with boundaries, sorry frontiers :-)

@TedShifrin so I do that thing where I only accept evidence that confirms my beliefs

if it doesn't confirm my beliefs, is it even evidence?

In my ponion, no

I only accept evidence that does not agree with me.

2:14 AM
he only accepts evidence that people considering accepting evidence would not themselves accept.

My beliefs are basically self confirmatory axioms,

contradictory axioms opens up a whole world of options...

you should talk to his barber.

Cogito ergo?
I dunno latin

i'm pink therefore i'm spam?

2:25 AM
Quod cogito ergo est?
I think it therfore it is?

si non caste tamen caute.

I think it therefore it is

cogito ergo sum

roughly, if you can't be good, be careful
my family motto.

my mum used to say, be good and if you can't be good, don't get caught

2:27 AM
I'm having a tough evening

sorry to hear it, geocalc. i hope the difficulties are purely mathematical.

in the 70's, irish police (the garda) were unbelievably connected, esp. in rural areas.
so not getting caught was important

my grandfather must have shot out every streetlight in south boston. the better to conceal his behavior.

Before I got summoned by theodore, i was thinking of a question re eigen vectors

is it a simple question that i can answer?

2:29 AM
Hey guys,
[interesting question, but I am not understanding what's exponential integrals](https://math.stackexchange.com/q/4109949/855081)

Any helps with this would be appreciated!
1

Consider the following quadratic program $$\begin{array}{ll} \underset{x_1,x_2,x_3}{\text{minimize}} & x_1 x_2 +\frac{1}{2} x_1^2 + \frac32 x_2^2 + 2 x_3^2 + 2 x_1 + x_2 + 3x_3\\ \text{subject to} & x_1 + x_2 + x_3 = 1\\ & x_1 - x_2=0\\ & x_1, x_2, x_3 \geq 0\end{array}$$ First, I want to check i...

I hope so, i wanted to know why it made sense to use eigen values / vectors in a piece of code that interprets an image as a connected graph (see the example on the wikipedia page on laplacian matrices)

not simple if it involves deciphering wiki pages

Can anyone help with it, understanding exponential integrals or the third solution by noname.

@copper.hat looks like the page has been edited since i last looked

2:32 AM
the top rated answer does not involve exponential integrals. i like it.

@leslietownes yes, but I am not familiar with what it involves..

Found the example : en.m.wikipedia.org/wiki/…
My question is, what does it mean to say transform into coordinates system of eigen vectors? And why does that work?

My evening was tough because someone I viewed as a friend viewed me as an acquaintance!

They take a vecotr of eigenvalus, from a laplacian matrix that represents the graph of an image, and multiply that by a vector of datavalues. The data vector being a single row/column, so when multiplied by the vector of eigenvalues produces a nxn matrix (where n is number of eignvalues)
@geocalc33 that hurts, esp when they are of the opposite gender (pr gender you are attracted to)
@geocalc33 i had a v. rough evening in my early /mid twenties like that. I got on my motorbike, and was riding soo slow because I was worried I might do something silly if i opened up the throttle
Ironiclly I had an accident a few days later, and when the police arrived to find me maniclly smiling they where all "and what is so funny, huh, are you high?"

2:47 AM
a quote from one of my favoUrist usa politicians, moynihan: “To be Irish is to know that in the end the world will break your heart.”

And then I had to sheepishly tell them about having a broken heart and existential thoughts, and that falling off my bike hurt so much that I was glad to be alive :)
@geocalc33 if you get a chance, I would highly recommend "essays in love" by Alain de Botton. Though it may not have aged well

To be Irish is to know that your existence is more or less equivalent to that of the potato.

we have sir walter Raleigh to thank for that...

Born in my county!
Didn't know that

some irony there (Raleigh bringing spuds to ireland)
some delicate history ensued :-)

3:16 AM
he is best known in american legal circles for his relation to the rule against hearsay.

that is educational!
potatoes were a gift to the queen apparently...

attorneys familiar with continental legal systems do not understand the rule against hearsay. bringing up sir walter raleigh does not improve the situation.

@leslietownes Sir I created a chatroom with you for discussing mathematics problems..It would be great if you would join..

see, one reason we do this is because there was this guy, yes, in the 1600s, something of a pirate, not even in the USA which didn't exist yet, but we're serious about this, ...

I have already sent you a request..

3:33 AM
:-)

there should be a synthetic geometry tag, no?

if this were 1921 and not 2021, yes. now, i'm not sure what that means.

geometry without algebra

oh, i see.

like, euclid stuff

3:37 AM
maybe axiomatic geometry. we could just call it geometry. if i'm being honest i don't see a lot of those posts. and they tend not to specify their axioms, which is a minus for me.

why do people care so much about having a tag?

I think synthetic geometry really is its own thing, it would be super useful for like people into competitive math

seems a bit niche to me/

i never understood competitive math. i realize it's a helpful laboratory for developing skills, i just never personally was exposed to it and it was weird to me.
when i started grad school someone asked me for my favorite 'synthetic geometry' result (not her words but very much the spirit of, euclidean geometry). i was like "uhh?" and she said something about her favorite. she dropped out in a year and coaches math students in competitions now. very effectively.
i just don't personally get it.

the problems, which challenging, always seem very contrived to me.

3:42 AM
well, those circle inversion theorems are excessively helpful for complex plane

knowing that a problem has an elementary result that someone else has already found is a huge leg up in problem solving. which isn't to say that problem solving skills aren't useful. i wish i had them. but, it's very much, as copper hat says, contrived.

yes, i do not wish to downplay the skill required to solve.

i never knew any math phds until i actually went to college. i may have benefited from earlier exposure to mathematics, including contest math. but i didn't.

imagine if you knew conformal maps are related to origin on the basis of thales' theorem. immediate intuition, bam, conformal map mastery

its a bit like f1 racing (which i love). very contrived.

3:46 AM
and suppose multiplication stopped working tomorrow
what do you do?
synthetic geometry!

$\bot \implies$ ???
i have my slide rule handy at all times.
crap, i need to move all four cars off the street for street sweeping tomorrow.
don't ask why 2 drivers have 4 cars

the ferrari f40
the bugatti
the volvo 240 for the 80s vibes
and his daily driver

i wish. albeit it would be an aston martin
or, if i had my personal mechanic, a red v12 e-type
co i can compensate in style
maybe a lancia delta hf integrale

you need one volvo.

my da's boss used to have a volvo and we salivated because it has electric windows
but we would not give his kids the satisfaction of using them even though we were dying to

3:55 AM
my wife had a 940 when we first moved in together. the electric windows were a kick.
i took my driving test in that car. the examiner demanded that i speed on city streets, and then took a call from her husband for about half of the driving exam. she wasn't paying attention to anything.
i slowed down for some workers on a highway. "why are you slowing down??" i don't know, because i'm one lane from human beings who would die if i hit them.

my fave car from my dad's colleagues was the citroen cx

i'd love to drive one of those.
or a morris minor

but i got to drive all manner of daimler, jaguar, volvo, landrover, etc. fun summer 'jobs'

do you think exams should be 'as much time as needed' within a 12 hour period to complete? like uni math exams
wouldn't that make results more representative of students ability

our gardener had a morris minor and the signals were physical things that extended out from between the doors.

4:00 AM
like in the real world people don't expect you to work within some two hour window of time you've been allocated
as long as its done by the end of your work day, you're usually good

in the real world there is always time pressure
unless you get to my stage of life

well yeah, i mean in the real 'math world'

interesting concept!

if you're a graduate student, and you're stuck on something but always get threw it in a 12 hour window
why does it matter if it took you 12 hours rather than 2?
*through

creative work can rarely be estimated in terms of time.
somethings you just accept and move on.
my mom has an nsu prinz. a truly awesome car. 500cc rear engine, air cooled.

4:02 AM
exactly, and if we dont want to shoehorn students into coming up with the 'model solution' to problems, wouldn't 12 hours allow those students who typically think of equally valid solutions that aren't the model solutions flourish more than students who are really good at internalizing the 'typical exam style' solutions
yeah sure, thats life

life is suboptimal
why porridge mathematics?

as an abstract proposition, time is a pretty bad measurement of work, labor, insight, anything. i work in an industry where we tend to bill by the hour. which means you're "penalized" (in a sense) for working efficiently.

i would change to my realname but there are a million people with my name

idk, i just tend to peruse maths while eating porridge in the morning

even today at the orthopaedist, two people with my exact name. wtf
i used to make & eat oatmeal every winter morning.

4:05 AM
thankfully my employers understand this, but it always comes up. i hit a minimum hours requirement and there's always a question why it isn't more. because i don't bill to kingdom come for something that takes me 20 minutes.

i love oatmeal

i eat oatmeal every morning.
uncooked, just with milk and blueberries.

i now hate oatmeal
so many freezing mornings waiting for the pot to boil.

:-)

4:06 AM
yeah, just don't cook it. get the pre-rolled steamed whatever. you can just eat it like cereal.

Just dump some oats in a pot with tasty stuff and stick it in the fridge overnight!

i'll stick with my cotswold on toast :-)

That's my favourite breakfast

i like irish/uk sausages

before i reached the point in my life when i could afford blueberries, i'd use raisins.

4:07 AM
mystery meat but tasty

Yeah
Cumberland
Toad in the hole with cumberland sausages
the dream
also scares people when you say the name

:-)
bangers & mash fine for me :-)

Also great
My go-to meals in a pub in the middle of moorland

i had a friend whose dad was a veterinarian and a butcher and used to make his own high quality sausages. but they were never as tasty as the mystery meat.

i think i'd take my animal to the veterinarian who wasn't also a butcher

4:10 AM
There's something alluring about miscellaneous trim

plus a fry :-)
chips & curry, and other delectable delights
i miss that stodge
its all healthy here in california.

Aye same, in Germany all we have is "good quality meat", whatever that is

my first time in germany i discovered the restaurants in chain stores. everywhere has them now, but they were amazing to me.
my friend worked in a hotel in mainz and i went to visit him

Ah that's not too far from me

i was surprised at how helpful people there were to me.

4:14 AM
Don't think that trait has survived the passage of time

i liked germany, especially the south
maybe i just looked particularly helpless.
when i grew up, ireland had a lot of german/dutch industrialists (today's billionaires) who were not terrible friendly to put it mildly.
which is why i was surprised

lol, the most helpful person I've met here so far was a Syrian refugee who worked in a bank, a shame that they get so much abuse.

unfortunately the world in general is not kind to strangers
in many cases there is good historical reason, but obviously the individual does not represent the group.
i used to go to london a bit with my da (he worked in the meat industry) and learned to keep my mouth shut (this was in ira times).

oops

i went back a few years ago with my kids and it was soo awesome not to have to care
no little green book at immigration
no pat downs
awesome.

4:18 AM
no bell bottoms, no fuses

plus i love the science & tech museum.
my brother just found a pair of my old flares

hopefully we can get right back to the firm border that england wants

as in trousers

the science & tech museum is amazing.

I've only been to London twice rofl

4:19 AM
i also love the hunterian museum. where else can you see a segment of charles babbage's brain.

the sensational press makes a big deal about ireland vs uk, but the reality is that from a practical standpoint they are much closer than most realise

the V&A is also amazing. i hope to take my daughter there some time.

for example, my daughter was able to vote in uk elections.

One time for an interview for a scholarship, the other time for a German exam. My uncle took me both times and both times we went out drinking afterwards and got absolutely slaughtered. Unsurprisingly I don't remember much about London.

on the basis of being an irish redident
i love london. some areas, like every big city, are best avoided, but it has some awesome stuff
new word, redident

4:21 AM
a friend of mine ran for MP for the lib dems in a district where she was guaranteed to be shellacked by some conservative, and she was. i don't know if anybody learned anything from that.

Ah I remember the Lib Dems
shame

lol he's back in the news

apparently he's 'poor'

4:23 AM
missed out on his 23 mil
:/

he should not have rolled the dice

does he have a gofundme? i'll donate 10 cents.

sorry if this seems vague but, does anyone else find you do maths better when you take a 'gentle' approach to thinking/problem solving?

he wont get a brass farthing from me

I'll donate a punch
@porridgemathematics it is vague lol

4:24 AM
i need to hit a crunch before the creative problem solving steps in

like when you think a little more 'lightly' things fall into place a little more smoothly

porridge that is super vague. i have never understood anything except by long exposure and getting used to. it's not understanding. no lightbulb goes on.

so i procrastinate until near deadline. sheer panic helps me focus

You think about it hard, cry a little, and then put down the pen and walk away

hmm, i guess everyone works differently

4:24 AM
that#s when the real work happens

a week ago tonight i was preparing a 28,000 word filing at the patent office. sheer panic helped me focus.

i forget soon however, which is why i keep rudin etc at my side.

I have that problem too, how do you get through education having a sub-par memory? I can understand the arguments and tend to remember objects and definitions, I just forget the trickery often used in proofs!

current list includes the pleasures of counting, something by marseden, rudin, dummit & foote, intro to minimax by some Russion fellows

the pleasures of counting is an amazing book.
my memory is absolute garbage for the names of results or where they come from. i can remember notions and vibes. they may not be captured in results.

4:27 AM
i the author is cambridge, i just knew it from the writing

tom koerner's companion to analysis is also a pretty good analysis book. i don't think it would adapt well to the united states but it must be a delight to teach from at cambridge.

i can remember some spooky detail (usually something to do with the opposite sex, wonder why) but can forget important stuff

i guess more concretely i am saying calm is a better state of mind for focus than panic, but i stand corrected since some answers here convey the opposite is true for them

i am thanked in the acknowledgments for one of his more recent books because i caught about 20 million typos.

learn to focus under panic, it is a useful tool.

4:28 AM
porridge i think that is true, for understanding and coming up with new results. for problem solving there is always a deadline and you never have enough time.

yeah thats true

at work when people explicitly ask me to do stuff on a deadline, the time frame can be anything from 5 minutes to 5 hours, but it is usually not longer than that. it is fast, and ugly, and not contemplative.
i don't propose this as a model for academic research, but it very much distinguishes academia from the chaotic soup that i live in.

one reality in the software/hardware world is that people always want something done by a deadline, but the deadline is often moved

thats fair enough, i should have been more specific and said i am really talking about focus for learning/research

Benny Hill "Just because nobody complains doesn't mean all parachutes are perfect."
most of my ideas came on bart

4:31 AM
for learning you want as much time as possible and as many books as possible. and to just sit back and absorb stuff like a sponge.
i had an idea for a paper come to me in a dream once. it was the weirdest thing. i woke up, and immediately wrote down what was in the dream, and then the paper was published. i do not promote mysticism but this did happen.

i learn best vicariously, particularly if someone is struggling with something

i dont think thats mysticism, people use hypnagogic states to their advantage to come up with ideas that are otherwise harder to
im sure a lot of people can relate to that, and its a well documented phenomenon

it was just very weird for me. i remember panicking to write down all of the stuff from the dream. i grabbed empty envelopes from junk mail on my desk to get it all on paper before i forgot it.

i found simple focus exercise/meditation helped me a lot

newton credited hypnagogia for a lot of his ideas

4:34 AM
i used to keep a notebook beside my bed. i don't have an good ideas anymore so it is not necessary
plus i sleep very little

it was so weird. the next meeting with my coauthor, i said, "we're done" and i showed him what i had seen in my dream. and we were done.
i get why people are religious, i'm not, but i get it. i can't explain that.

yeah that is pretty amazing

most of my dreams now have to do with stuff that is funny, i wake up laughing and then i can't remember the joke.

a whole paper

like one of my advisors said, i think religion is a good thing... for other people.

4:43 AM
my mom's become one of those people who goes to church more than just on sundays. her priest, who is irish, loves it. no criticism, just saying.

it used to be the internet for ireland, that's how news spread. morning mass and news exchange. facebook had nothing on irish gossip

there is a substitute priest who is nigerian and also loves it. i think they've formed some kind of conspiracy to get my mom in there. this is beyond the scope of a math chat.

#bot startup-process8696=START#
Hi

irish gossip dot com would be quite the startup. i would submit 50 million. if i had it.

:-). never quite capitalised on my ethnicity

4:47 AM
now's the time. i suggest investing in a green suit and posing next to a kettle or 'pot' if you will that appears to be full of golden coins.

:-). i was a little too unpredictable to the 'i love your accent' crowd.

Religion may fill the human need for finding answers to existential questions and why ee die and suffer
I am myself religious

i did make a recording a couple of weeks ago for a 7yo where i pretended to be a leprechaun

"pretended" to be.
(narrator: he is a leprechaun)

i think it is an artifact of some part of us that has to be evolved out.
we have gone from multiple gods to one god. only a matter of time.

4:50 AM
laughed out loud. although not laughing at my mother.

i like the broad humanist aspects.
i like the social aspects.
i hate the mind bending. because my mind doesn't bend easily :-)

i really liked free donuts outside of st. john's when i was a kid.
we moved to st. thomas aquinas and they had a different donut policy. this convinced me that there was no god.

i liked watching the girls in their sunday best

Wow a very nice reason for believing there is no god

i just felt that a god would give me donuts on sunday. if i'm wrong, i'm wrong.

4:52 AM
i always thought that they should charge for communion and offer flavoUrs
and you could pay for less penance.
i was a bit behind the times there with indulgences

:) i don't mean to disparage any religious tradition. i just think that st thomas aquinas was unusually stingy regarding their donut budget. and that is a secular concern that probably does not embrace the almighty.

@Euler2 i apologise, i do not wish to annoy or aggravate you.

Well i don't get annoyed by anyone
Everyone has there own beliefs

hmmm...

that's all i want

4:54 AM
i also do not understand the power music holds over people.
i do not dispute it. i just do not understand.

my wife was trying to get me to go to a church and they had people plugging in at that church. like, electric guitars. that's the same crap i do in my office.
if we want to listen to that we can do that at home.

i do like some older church music.
as would be fitting in a catherdral.

if i'm going to church i want to hear choral music. nobody is plugging into anything.
the only reason i plug my electric guitar into an amp is to summon satan himself. so don't talk to me about this stuff.

bill gates?
or balmer?

4:58 AM
i've got 0.5 of the bill gates 5g corona experience coursing through my veins. to be honest, it hasn't been a pleasant ride. my arm hurts, has been constantly tingling, and i've been feverish to 101 degrees.

wow. sry to hear it.

but the fever has gone and now i'm just a guy with a sore arm.

call a dr
ok.

i've emailed a bunch of people tomorrow, hey, i might be going to the doctor instead of getting back to this.

@leslietownes Krispy Kreme will give you a free one for being vaccinated!

4:59 AM
but my temp is normal and my arm is not as bad as it was 12 hours ago.
i have yet to obtain my donut.

have a pint

i drew a shamrock in the foam.

used to be if you donated in ireland you got a pint

first shot. a guy at my work said, if you have been infected, the first shot might be bad. i don't know if that's true. i had a period last october when i couldn't eat for four days. it might have been this.

5:01 AM
sacrilege
get a test.

i tested negative when i last got the test.

study harder

i'm just happy to not be at fever temperatures.
WORK HARDER
that's what i yell at my toddler when she doesn't complete the puzzle on time.

coudl be unrelated, or maybe they infected you. or...
i know how that works. i am sure she rules the roost

I had fever and headache after the second, but I've heard that theory before. Hang in there.

5:04 AM
covid is frustrating

it is also killing my investments

im a bit confused by why in the definition of a 'countably normed space' we need the norms to be pairwise compatible, in the sense that if a sequence is cauchy in both norms and convergent in one, it is convergent in the other - surely if we just had a countable sequence of norms all on a linear space, the system of neighbourhoods $$U_{\tau,\epsilon} = \{x \in V : ||x_i|| < \epsilon, i=0... \tau \}$$ is a local base at $$0$$ for the topology induced by
$$\rho (x,y) = \sum_{n \in \mathbb{N}} 2^{-n} \frac{ ||x-y||_{n}}{1 + ||x-y||_{n}}$$ regardless of whether the sequence of norms is pairwise compatible?
so is the upshot of requiring the norms be pairwise compatible that as long as one of the norms is cauchy, then the topology induced by $\rho$ is complete?

are they seminorms?

no bonafide norms

$\|x\|_i$ presumably.

5:18 AM
oh yeah, thats a mistake, $||x||_{i}$

#bot startup-process8697=LEAVE#

@Euler2 what does that mean?

5:36 AM
@copper.hat don't you know that I am a bot

hmmm

13 hours ago, by leslie townes
none of you are people, you're all bots, you're my CPU come to life. F- on the turing test.

if I have a sequence of schwarz functions $\{f_p \}_{p \geq 1}$, and for some fixed $0 \leq m < n$ I know $f_p$ is cauchy in the norm $||f||_{s} = \sup_{t \in (-\infty,s) ; k,q \leq s } |t^k f^(q)(t)|$for $s=m,n$, and $f_p \rightarrow f$ where $f$ is schwarz, in $|| . ||_{m}$ , how can I show it converges to $f$ in $|| . ||_{n}$?
I feel like I can't and the definiton of these norms is wrong, rather it should be a supremum over the reals.. not from $-\infty$ to $s$
oh no, the notation looks bad, the norm should read $\sup_{t \in (-\infty,s) ; k,q \leq s } |t^k f^{(q)}(t)|$
as in $q$th derivative

if you mean Schwartz, then yes, it is over the reals.

so it shouldn't be only up till $(-\infty,s)$ for each norm indexed by $s$?
and would what I am asking be false in that case
yes schwartz

5:46 AM
i don't know what you are looking at so i can't answer, but typically one would have $t \in \mathbb{R}$.

they're my uni courser notes, not from a textbook
yes I see
I guess another way of phrasing my issue is if we know that $f_n^{(q)} \rightarrow 0$ uniformly on $(-\infty, m)$ for some fixed positive integer $m$, and for all $q = 0...T$ for some fixed positive integer $T$, and we know that $f_n^{(q+1)}$ is uniformly cauchy on $[m,m+1)$, can we say for sure that $f_n^{(q+1)} \rightarrow 0$ uniformly on $(-\infty,m+1)$?
we may also assume $f_n$ is schwartz, im thinking because its such a nice function, this could be true
but it doesnt seem easy to prove

1 hour later…
7:03 AM
Maybe somebody well versed in measure theory would be willing to have a look:
in Modern Abstract Analysis, yesterday, by Alphie
Hello everyone this is my first time in a chat room on Math SE. Am reading the proof of Fubini's Theorem in Lang's book real and functional analysis and I have some questions. Anybody familiar with this book?
in Modern Abstract Analysis, 17 hours ago, by Alphie
My question is why do we need to invoke Corollary 5.10 on page 163 to show that f_x is integrable? It seems to me that integrability follows from the definition of the integral since we have an L^1 Cauchy sequence of step maps converging the f_x for almost every x.

1 hour later…
8:06 AM
if a function is $C^1$ on $\mathbb{R}$ except at countably many points, is it absolutely continuous?

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