Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue-measurable function such that $f(x) \geq 0$ for all $f.$ If $$\int_{-\infty}^\infty f(x) dx < C,$$ for $C < \infty$ a fixed constant, why does there have to be a sequence $\{x_i\}$ such that $x_i$ goes to infinity, and $f(x_i)+f(-x_i) \to 0$ as $i \to \infty?$
If $f$ is continuous, this is clear.
If such a sequence doesn't exist, it means that there exist $\delta>0$ and $R\geq0$ with $f(x)+f(-x)\geq\delta$ for all $x\geq R$. Then $$ \int_{\mathbb R}f=\int_{[0,\infty)}f(x)+f(-x)\geq\int_{[R,\infty)}f(x)+f(-x)\geq\delta\,m([R,\infty))=\infty. $$