# Slices of measurable functions are integrable?

1. Oct 22, 2011

### Kreizhn

1. The problem statement, all variables and given/known data
Let f and g be two measurable functions on $\mathbb R^d$. Show that their convolution $(f \star g)(x) = \int_{\mathbb R^d} f(x-y)g(y) dy$ is well-defined.

3. The attempt at a solution

All that needs to be done is to show that for almost every fixed $x \in \mathbb R^d$ that f(x-y)g(y) is integrable. Now apparently this is almost trivial, since I have previously come across the statement

This is not obvious to me. Should it be? It is easy to show that f(x-y)g(y) is a measurable function on $\mathbb R^{2d}$ but it's not clear to me why a "slice" of a measurable function need be integrable.

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