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
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Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $14.8$
