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Suppose that $f\in \mathcal{L}(\mathbb{R})$ and $g\in \mathcal{L}(\mathbb{R})$, $\phi(x,y)=f(y-x)g(x)$, prove that $\phi$ is measurable in $\mathbb{R}^2$.

I try to prove this problem by the definition of measurable function:

For any $\alpha\in\mathbb{R}$, $E=\{(x,y)\in\mathbb{R}^2|\phi(x,y)<\alpha\}$ is measurable in $\mathbb{R^2}$.But I failed.

Are there some better ideas to solve this problem about measurable function in $\mathbb{R^2}$ ?

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    Hewitt & Stromberg, REAL AND ABSTRACT ANALYSIS, have this result. See (21.31). For the Lebesgue-measurable case it is about half a page. The main thing to be proved: If $\phi(x,y) = x-y$, and $M \subseteq \mathbb R$ has measure zero in the line, then $\phi^{-1}(M) = \{(x,y)\colon x-y \in M\}$ is Lebesgue measurable in the plane. It is not enough to use sigma-algebra Lebesgue x Lebesgue in the plane: you have to use the *completion* of that sigma-algebra.2012-05-18

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$\newcommand{\R}{\Bbb R}\newcommand{\Z}{\Bbb Z}\newcommand{\N}{\Bbb N}\newcommand{\Q}{\Bbb Q}$ I don't know how many tools you have available, but a solution can go like this.

If $F$ and $G$ are measurable functions defined in $\R^d$ then $FG$ is a measurable function. A proof of this fact can be found here.

Then it's enough to show that the maps $(x,y)\mapsto f(y-x),\qquad (x,y)\mapsto g(x)$ are measurable functions.

First, we need the following Lemma.

Lemma. Let $E\subseteq \R$ a measurable set. If $[a,b[$ is a finite interval, then $E\times [a,b[$ and $[a,b[\times E$ are measurables in $\R^{2}$.

Notice that it's enough to prove the Lemma for the interval $[0,1[$. Then the proof of the Lemma is in stages:

  1. First consider $E$ with $m(E)=0$
  2. Then consider $E=(c,d)$ a finite open interval.
  3. $E$ open set with finite measure.
  4. $E$ a $G_\delta$ set with finite measure.
  5. $E$ a measurable set with finite measure.
  6. $E$ a measurable set.

Notice that from the Lemma follows:

Corollary. If $E\subseteq \R$ is a measurable set, then $E\times \R$ and $\R\times E$ are a measurable sets of $\R^2$.

To prove the Corollary, note that $E\times\R=\bigcup_{n\in\Z} E\times [n,n+1[.$

Finally

Lemma. Let $f:\R\to\R$ a measurable function. Then the functions $F,G:\R^2\to\R$ defined by $F(x,y)=f(y),\qquad G(x,y)=f(x)$ are measurable.

Proof. Let $\alpha\in\R$. Notice that $\lbrace (x,y):F(x,y)\lt\alpha \}=\R\times \lbrace y:f(y)\lt\alpha\},$ by the corollary above, we are done.

To finish note that $\lbrace (x,y)\in\Q^2: f(y-x)\lt\alpha\}=\bigcup_{x\in\Q}\lbrace x\}\times(\lbrace y\in\Q : f(y)\lt \alpha\}+x),$ is measurable, therefore $(x,y)\mapsto f(y-x)$ is measurable.

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    I already figured the proof in Stein using Tonelli's theorem, I was just looking for a more elementary proof. Thanks !2013-12-02
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First of all, the $\sigma$-algebras $\mathcal{B}(\mathbb{R}^2)$ and $\mathcal{B}(\mathbb{R}) \otimes \mathcal{B}(\mathbb{R})$ are the same, since $\mathbb{R}$ is a separable metric space.

So, a function which takes values in $\mathbb{R}^2$ is measurable if and only if its two coordinates functions (which take values in $\mathbb{R}$) are measurable.

The map $(x,y) \in \mathbb{R}^2 \to x - y \in \mathbb{R}$ is continuous, and so is also measurable.

Hence, the map $(x,y) \to f(x-y)$ is measurable (by composition). The map $(x,y) \to g(y)$ is measurable (composition of the second coordinate map with $g$). We can deduce that the map $(x,y) \to (f(x-y),g(y))$ is measurable.

Since the map $(x,y) \to xy$ is measurable (because continuous), by composition we have that $(x,y) \to f(x-y)g(y)$ is measurable.

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    I haven't written out all the details of that argument in a while. It may be relevant that both mappings take sets of measure zero to sets of measure zero.2012-05-18