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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).

 

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