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Slices of measurable functions are integrable?

  1. Oct 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Let f and g be two measurable functions on [itex] \mathbb R^d[/itex]. Show that their convolution [itex] (f \star g)(x) = \int_{\mathbb R^d} f(x-y)g(y) dy [/itex] is well-defined.

    3. The attempt at a solution

    All that needs to be done is to show that for almost every fixed [itex] x \in \mathbb R^d[/itex] 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 [itex] \mathbb R^{2d} [/itex] but it's not clear to me why a "slice" of a measurable function need be integrable.
  2. jcsd
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