Garrett Black

2022-06-22

We have the notion of Lebesgue measure, which I generally think of as the length/area of some interval in the space considered. We also have the notion of Lebesgue measurable functions. I was wondering if there is some way to compute the Lebesgue measure of a function? Perhaps, for example the function $f\left(x\right)=\mathrm{sin}\left(x\right)$. The only thing that comes to my mind would be the arclength of the curve over some interval. Alternatively, can we describe the Lebesgue measure of a function as the length of the interval of the inverse image of the function?

odmeravan5c

If you say that a Lebesgue measure is a length/area of some sorts, then it seems you're thinking of original arguments being sets. That is $\lambda \left(A\right)$ should give some number, which you understand as a measure of $A$. In that case, your question "what is a Lebesgues measure of a function" to me immediately triggers the concept $\lambda \left(f\right)$ which is simply an integral: $\lambda \left(f\right)=\int f\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\lambda$.
Note that those two concepts are essentially the one. Namely, if you define (some) measure $\mu$ for sets, you can get its action on functions starting by approximating them with simple functions. On the other hand, if you managed first to define $\mu$ for functions, then you get its actions on sets through the indicator functions: a measure of a set $A$ can be obtained by $\mu \left({1}_{A}\right)$.
I would not say that a pushforward measure is a measure of a function, as it is rather a change of measure using a function (or a map in general).

Zion Wheeler

Recall that measurability of a function is not defined in the same way as measurability of a set. You define measurability of a function to be able to integrate it, and so your "measure" of $f$ would most naturally be its integral, while you define measurability of a set to be able to measure it using a measure. So if you want some notion of the measure of a function $f$ on some measure space $\left(X,\mathcal{A},\mu \right)$, the most natural notion would simply be
${\int }_{X}f\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\mu .$
I'd also like to say that this makes a lot of sense if you look back at the basic definitions though simple functions, where you have
${\int }_{X}{a}_{j}{\chi }_{{A}_{j}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\mu =\sum _{j=1}^{n}{a}_{j}\mu \left({A}_{j}\right)$
showing you even more explicitly how the function is "measured".

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