Convolution of Integrable Function with Bounded Function

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Theorem

Let $f: \R^n \to \R$ be a Lebesgue integrable function.

Let $g: \R^n \to \R$ be an essentially bounded function under Lebesgue measure $\lambda^n$.


Then the convolution $f * g$ of $f$ and $g$ is bounded and continuous.

In particular, $f * g$ is again essentially bounded.


Proof


Sources