How are measures of "nice" subsets of $\Bbb R^n$ defined? For example in physics one often integrates of the energy levels $H^{-1}(E)$ of some smooth function $H$. Specifically right now I am interested in getting a measure on the boundary of a (bounded) convex set.
(lebesgue measure is the only thing that exists as an inducing measure here)
@Alessandro does this measure agree with the induced measure you get when your subset is a manifold and you look at the measure you get from the induced riemannian metric?
@s.harp: If you're working with something other than an open set of $\Bbb R^n$, Lebesgue measure won't be good. You either have to use its manifold structure or use Hausdorff measure of the appropriate dimension.
Well, it's topologically a circle, as Leaky just showed, and as a group it is the circle group, as well. Addition in $\Bbb R/\Bbb Z$ corresponds to multiplication in $S$.
@BalarkaSen "And when his work is done, haha! begins the fun--from Dnepropetrovsk to Petropavlovsk by way of Iliysk and Novorossiysk to Alexandrovsk to Akmolinsk to Tomsk to Omsk to Pinsk to Minsk to meeee the news will run!"
@TedShifrin My research is delving into numerically solving stochastic ODEs, which is...to put it lightly, not to my tastes, but hopefully it can get me a paper. Outside of that, I have a class in mathematical models (...) and a class on the Lebesgue integral (!!!).
Oh, that reminds me, @Fargle. I have more than a dozen recs to do for a former student/advisee who's applying to grad schools. I think my days of doing this are coming to an end soon.
@Fargle I have a decreasing sequence of measurable sets $E_n$, such that $\bigcap E_n=E$. I'm trying to work out conditions under which $\lim\limits_{n\to\infty}\int_{E_n}f(x)\mathrm{d}x=\int_E f(x)\mathrm{d}x$
I think I can prove that this holds if $f$ is nonnegative on $E_1$ and $\int_{E_1}f\mathrm{d}x<\infty$, because in this case the function $A\mapsto\int_Af\mathrm{d}x$ is an outer measure and this becomes the continuity theorem for outer measures
