Reference for: a Lebesgue integrable function and its rearrangements have the same Lebesgue integral.

Question

In http://kunklet.people.cofc.edu/condint.pdf there's the sentence at the beginning of the Introduction:

"An elementary fact from measure theory states that a Lebesgue integrable function and its rearrangements have the same Lebesgue integral."

Can someone please refer a book that proves this statement? Thanks.

Answer 1

Denote $f_\#\mu=\mu\circ f^{-1}$, namely the law of $f$ under the Lebesgue measure. Since $f$ and $g$ have the same distribution it follows that $f_\#\mu=g_\#\mu$ (uniqueness of measure). Moreover one has by the change of measure/variables formula $$\int_\mathbb R f(x) \, d\mu(x) =\int_\mathbb R x\, df_\#\mu(x) $$ Using $f_\#\mu=g_\#\mu$ we conclude that $$\int f\, d\mu=\int g\, d\mu$$ Surely, you have to verify why the uniqueness of measure theorem applies.

Answer 2

Use the definition in the link. Let a strip be defined by $[a,b)$. Then from the definition of rearrangement the measure of the set $(x:a\le f(x)\lt b)$ is the same as the measure of the set $(x:a\le g(x)\lt b)$. Therefore the strip approach leads to the integrals being equal. In case it isn't obvious, use the fact that the sets $(x:f(x)\lt b)-(x:f(x)\lt a)=(:a\le f(x)\lt b)$ so the rearrangement definition applies to strips.