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## Useful continuity

Suppose $f: \Bbb{R}^p \to \Bbb{R}$ is integrable and bounded. Prove that the mapping
$(\Bbb{R}^p)^n \ni (a_1,a_2,...,a_n) \mapsto \displaystyle \int_{\Bbb{R}^p}f(x+a_1)f(x+a_2)...f(x+a_n)f(x)dx$ is continuous.

Then use this result to solve the following problem

Proof: In all considerations below we work with the Lebesgue measure, and we denote by $m(X)$ the measure of the Lebesgue measurable set $X$. First of all, let’s prove that our function is well defined, which means that the integral is finite. To do this, first we show that $L^q \subset L^1 \cap L^\infty,\ \forall q \in (1,\infty)$. First, if we denote $A=\{x \in \Bbb{R}^p : f(x)\geq 1\}$, then this set has finite measure. Indeed, $m(A)=\int_A 1 dm\leq \int_A f dm \leq \|f\|_1$. Pick $q>1$ then $\int_{\Bbb{R}^p}| f|^q dm=\leq \int_A| f|^q dm+\int_{\Bbb{R}^p\setminus A}|f| dm \leq m(A) \|f\|_\infty^q+\|f\|_1<\infty$.

Denote $g_k(x)=f(x+a_k),\ k=1..n$. Then $g_k \in L^1 \cap L^\infty$, which means, by the previous argument that $f,g_k \in L^{n+1},\ k=1..n$. See this post to conclude that $g_1g_2...g_nf \in L^1$, and therefore the integral is finite.

The next step is to approximate $f$ with a continuous function with compact support $h$. First we prove the condition for such function, and then we use the density of the space of continuous functions with compact support in $L^1$ (see density. Source http://www.math.pitt.edu/~frank/2300/density.pdf).