Thank you for the advice :)

So the closed unit ball in $\ell^2$ isn't precompact, but it has the weaker property that every uniformly continuous function into $\mathbb{R}$ is bounded

and I am wondering, how is this property called, if anything. A name I could make is weakly precompact

maybe weakly precompact is a bad nomenclature considering how precompact is used in the context of vector spaces

maths.tcd.ie/pub/ims/bull53/R5301.pdf

apparently people did study this for metric spaces

apparently this is equivalent to, for each $\varepsilon > 0$, we can decompose $X$ into finite amount of sets $X_1, ..., X_n$ such that for every $x, y\in X_i$ there exists $x_1 = x, x_2, ..., x_n = y$ with $d(x_i, x_{i+1})\leq \varepsilon$ and we can do so while keeping $n$ bounded

sorry, replace $x_n$ by $x_m$ and last $n$ by $m$

so it is pretty much like totally bounded, but that one has $m = 1$

whereas here we allow longer steps

and by bounded $m$ I just mean, lets say, $m = m(\varepsilon)$ is a fixed natural depending on $\varepsilon$

@AlessandroCodenotti @Thorgott look at this pretty cool weakening of totally bounded metric spaces I've found

might generalize this result to uniform spaces

The new Disney movie, Wish, apparently was a big flop

plot, animation style, moreover it was 100th movie anniversary

apparently everything about the movie had a drop in quality in comparison to the other films

did any folks here ever read feller (An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition) for probability?

(Also if anyone gets the chance to look at my question above i would be very grateful :) )

ee18 feller is a classic (maybe not the best if you also want to learn the most modern mathematical frameworks as quickly as possible)

what do you mean by mathematical frameworks here Leslie?

as in measure-theoretic probability?

i assume no seeing as that's not modern :)

well actually i did mean anything to do with abstract measure theory, which as i recall feller either does not do or doesn't do until late

If you do have any other probability suggestions for someone after a first course in analysis (still months to even a year away to be fair) i'd also be all ears

or anything else that you'd need to do to rigorously develop basic theory of stochastic processes in continuous time

ah that is more problematic

i don't have other suggestions, i'm just saying, if you pick up feller i hope you actually want to learn probability theory and not measure theory

oh for sure, i definitely do

if you want to get to like black-scholes and stochastic DE as quickly as possible, there was some text that the financial math people i knew 15 years ago really liked for that that wasn't feller, but i don't remember what it was

Rosenthal?

maybe not for math math people actually

i might be thinking of evans

Probability and Statistics: The Science of Uncertainty?

evans 'introduction to stochastic differential equations.' it was a circulated set of notes before it was a book

it's kind of the opposite though, it assumed you know at least a little measure theory

ah ok, so it might be a feller followed by evans sort of thing?

you're saying evans might pick up what modern stuff feller misses?

if you want to do stochastic differential equations, maybe start with some book that has that more in mind than feller

feller's first few chapters are balls and urns and shit

i've heard it reads like a good story though ;)

i mean it all comes down to what do you want from a book

haha

fair play to that

hopefully i'll have a better answer to that once i have to decide on a book (months away)

the probability book i used five years ago was an engineering-ish one by papoulis, didn't love

i really like feller i just don't know that it's a great guide to what is going on right now in whatever is hot in mathematical applications of probability theory

i feel like a more modern book would get to abstract measure spaces sooner, even if they weren't emphasizing them

depends on the applications

I doubt there are people who do probability theory applications without any measure theory nowadays

physics? so i guess SDE probably matter there too?

well, SDE's are of interest in some physics, yes

as much as you can pick a field of math and say its of interest in some physics, I suppose

its probably almost all of them

better way would be to study math when you actually need it

if you want to absorb enough probability theory to address all potential applications, you might be in the market for more than one book

@EE18 whats your goal

is it just to learn some math

I assume not

This is all just for fun for me on the sides. To know more math and physics, i guess

I got fairly far with physics self-study over the course of 2-3 years but found myself not quite well-equipped enough with math from my engineering undergrad. ergo math time...and it turns out that's a ton of fun too. hence "wasting" time with logic and set theory ;)

this goes back to, you can spend ages learning just one field

for sure, and that's ok because in the end im not doing any of this professionally :)

learning logic and set theory is worthless in the first place for non-math applications

or at least for physics applications

unless you want to be professional goofball

LOL

definitely worthless for those applications, but a fun stretch of the mind. and maybe ill understand some of GEB next time i read it...

you want to start learning basic probability before even wondering about learning things like SDE

I've learned (and forgotten) basic probability theory to be fair

what that means to you and to me is probably different

I mean probability with measure theory background

they re-opened the pool near our house after several months, and my daughter and i went to swim in it, and this duck spent like 20 minutes next to the pool, just staring at us, clearly wondering when we were going to leave

ah that's fair jakobian

The duck was unimpressed with the company it seems

we used to swim with the ducks, i have a photo of my daughter with three ducks in the pool

this duck was clearly a newcomer who thought we were newcomers

I don't mean classical probability theory you learn in high school, like $P(A) = |A|/|\Omega|$. That's way more simpler

"you learn in high school" muffled laughter

they don't learn that in American high schools?

@leslietownes i've never had this said to me in lecture but i can only imagine the stereotype of the epic soviet math prof who...actually probably learned some of this stuff in high school

not in mine :)

jakobian, america does not have anything resembling a national educational system, high schools are controlled locally (~ 15000 districts) subject to state standards (which are variable among the 50 states)

there wasn't a whiff of probability at my high school but one county over in the rich area they may well have done all of it

it wouldn't surprise me if the rudiments of probability, as expressed in your example formula, are in at least some state standards

which isn't to say that local districts have ever been equipped to meet them

hmm. Weird. This is the simplest probability there is, and in Poland its usually what students score points on on their finals

i think my favorite activity is searching for a good book in field X and seeing the hilarious suggestions (made more hilarious by the fact they're not satire) for questions about "best intro to probability?"

Someone answers "Probability and measure, Billingsley". I've never read it but i know enough to know that is absurd

"and measure" tells you all you need to know

"billingsley" also sounds like the surname of someone one social class better than the reader

@EE18 well yeah thats a bit of a full introduction

@leslietownes LOL

AFAIK it's quite a famous book in probability circles?

at least, insofar as i had heard of it...

but if you're willing, you will learn a lot from it

if you're just trying to learn math, then why not?

i have about 6-8 months to go with my analysis, LA, set theory trifecta before getting to choose what's next but who knows, maybe billingsley will be it

okay, your majesty

would have made a better king than charles no doubt

well, which charles

the current monarch

@EE18 I suspect this won't go far. Learning three subjects at once like that usually doesn't make you progress. You end up lacking focus and thus compromising

is there another lol?

i wouldn't otherwise do it jakobian, and would otherwise drop set theory, but i am close enough to the end that ill push through. from there it will be just LA and analysis

ee18 he is charles the third, charles the first was executed for being kind of a dork

charles ii was OK in my personal ranking of monarchs

did not know we had an expert on the monarchy in our midst

i won't pretend to know anything more than the crown has taught me

the problem with ranking monarchs is they're all basically horrible

@Jakobian Ok, but thank you for your time!

absolute power does it historically, but i guess recently, without the power, there's less of an excuse. i do wonder if they'll still be around when i die or if the UK will become a republic...who knows

the problem with monarchy is that they keep it all in the family

you expect a genealogical tree? well how about a cycle instead

@EE18 my strategy would be to dump set theory, learn LA, then go for analysis

Can someone please tell me whether this is a proof by contrapositive? I don't understand the contradiction here.