Take a Hamel basis of $\mathbb{R}$ and define a map $f\colon\mathbb{R}\rightarrow\mathbb{R}$ which sends each real number to the sum of the basis elements of which it has non-zero component. Is there a continuous $g\colon\mathbb{R}\rightarrow\mathbb{R}$ such that $f=g$ a.e.?
@Thorgott no. If $f=g$ a.e., then $f-g$ vanishes outside a null-set, so in particular $f-g$ is measurable, this implies that $f$ is measurable. But there's a theorem that under some assumptions (e.g. locally compact Polish group to Polish group) a measurable homomorphism is continuous. Evidently $f$ is discontinuous
Lebesgue-measurable means the preimage of every Borel set is Lebesgue-measurable- Borel-measurable means the preimage of every Borel set is a Borel set, which is stronger.
Yeah, changing stuff in the codomain would ruin everything (the preimage of a Lebesgue-measurable set under a continuous map need not be Lebesgue-measurable).
@Thorgott the author explicitly writes before the theorem "Steinhaus eventually proved that any Lebesgue measurable solution is continuous and with the advent of abstract harmonic analysis in the 1930’s, this was extended by Weil to all locally compact Polish groups."
My second idea would've been some weird q-analog like $\dfrac{x^{1-q}-1}{1-q}$, something that becomes the usual log as $q\to1$, but that doesn't work in context
It only occurred to me recently, in connection with this MO posting, that the way I write the base of the logarithm is not shared by the rest of the world. I am Dutch, and I learned at school to write the base as a superscript, $^{a}\!\log b$, rather than as a subscript, $\log_a b$. Searching the...
Like what I'm doing is, starting at the center, going to the edge ($1$), going 3/4 around the circle ($\frac34(2\pi)=\frac32\pi$), and then going one unit away ($1$)
@TedShifrin I think you just need to hit the shore
@Ultradark Suppose you get lost on your boat at sea. You have a device that indicates your distance from coast is 1km, but you have no idea the direction you should head to get to the shore. You know also that the coast is linear, being perpendicular to the line between your location and the coast itself. Q: what is the minimal length of a path which guarantees you meet the coast?
