I am having trouble with an exercise within the book "Modern Real Analysis" by Ziemer.The exercise appears in chapter 6 (on integration) within section 6.8 on the dual space of $L^{p}$. The exercise statement is :
Let $f$ be a Lebesgue measurable function on $[0,1]$ and $Q := [0,1] \times [0,1]$.a.) Show that $F(x,y) := f(x) - f(y)$ is measurable with respect to the Lebesgue measurein $\mathbb{R}^{2}$.b.) If $F \in L^{1}(Q)$, show that $f \in L^{1}([0,1])$.
Here I assume the shared co-domain of $f$ and $F$ is the extended real numbers $\overline{\mathbb{R}}$. The measurable space defined on $\overline{\mathbb{R}}$ is$(\overline{\mathbb{R}},\mathcal{B}(\overline{\mathbb{R}}))$ with$\mathcal{B}(\overline{\mathbb{R}})$ denoting the Borel sets. $\lambda$ denotes the Lebesgue measure on $[0,1]$ or $Q$ depending on the context.
My solution so far is below :
a.) Let :\begin{align*}F_{1}(x,y) & = f(x) \\F_{2}(x,y) & = f(y)\end{align*}So $F = F_{1} - F_{2}$. Since sums of measurable functions are measurable and the measurability of a function is not changed when it is multiplied by $g \equiv -1$ :\begin{equation*}F_{1} \text{ and } F_{2} \text{ are } \lambda\text{-measurable} \Rightarrow F \text{ is } \lambda\text{-measurable}\end{equation*}Let $D \in \mathcal{B}(\overline{\mathbb{R}})$. We see :\begin{equation*}F_{1}^{-1}(D) = f^{-1}(D) \times [0,1]\end{equation*}where $f^{-1}(D)$ and $[0,1]$ are $\lambda$-measurable. According to ( Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?)$f^{-1}(D) \times [0,1]$ is also $\lambda$-measurable. So $F_{1}^{-1}(D)$ is measurable. Can use the same argument for $F_{2}$ so that $F$ is $\lambda$-measurable.
I still haven't been able to find a good approach for solving part (b.). Can someone help with this ?
