An inequality related to a Borel measurable function

Question

Let $f: (-\infty, \infty) \rightarrow [0, \infty)$ be a Borel measurable function. Prove that $$\int\limits_{-\infty}^{\infty}f(x)e^{-\pi x^2}dx \le Hf(0),$$ where $Hf$ is the Hardy-Littlewood maximal function.

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Answer 1

Hint: Suppose $g= \sum_{k=1}^{n}c_k\cdot \chi_{I_k},$ where each $I_k$ is an interval centered at $0,$ and $\int_{\mathbb R} g = 1.$ Can you show $\int_{\mathbb R} fg \le Hf(0)?$