# Convolution of Integrable Function with Bounded Function

## 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.