We have the notion of Lebesgue measure, which I generally think of as the length/area...

Garrett Black

Garrett Black

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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 ( x ) = sin ( x ). 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?

Answer & Explanation



Beginner2022-06-23Added 20 answers

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 λ ( A ) 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 λ ( f ) which is simply an integral: λ ( f ) = f d λ.
Note that those two concepts are essentially the one. Namely, if you define (some) measure μ 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 μ for functions, then you get its actions on sets through the indicator functions: a measure of a set A can be obtained by μ ( 1 A ).
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

Zion Wheeler

Beginner2022-06-24Added 11 answers

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 ( X , A , μ ), the most natural notion would simply be
X f d μ .
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
X a j χ A j d μ = j = 1 n a j μ ( A j )
showing you even more explicitly how the function is "measured".

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