We have the notion of Lebesgue measure, which I generally think of as the length/area...
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 . 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 should give some number, which you understand as a measure of . In that case, your question "what is a Lebesgues measure of a function" to me immediately triggers the concept which is simply an integral: . Note that those two concepts are essentially the one. Namely, if you define (some) measure 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 for functions, then you get its actions on sets through the indicator functions: a measure of a set can be obtained by . 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).
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 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 on some measure space , the most natural notion would simply be
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
showing you even more explicitly how the function is "measured".
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