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?

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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?

odmeravan5c · Accepted Answer

If you say that a Lebesgue measure is a length/area of some sorts, then it seems you&#039;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 &quot;what is a Lebesgues measure of a function&quot; 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 · Accepted Answer

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 &quot;measure&quot; 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&#039;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 &quot;measured&quot;.

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

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