Let $\Sigma$ denote the set of Lebesgue-measurable subsets of $\mathbb{R}$, and $m$ the Lebesgue measure on $\mathbb{R}$. Let $1<p\leq \infty$, $f\in L^1(\mathbb{R},\Sigma,m)$, and $g\in L^p(\mathbb{R},\Sigma,m)$. Show that
$$\int_{\mathbb{R}}f(y)g(x-y)\mathrm{d}m(y)$$ exists for almost every $x\in \mathbb{R}$.
I can only assume that they mean that the above integral is finite almost everywhere. So, I must show that $$\int_{\mathbb{R}}|f(y)g(x-y)|\mathrm{d}m(y)<\infty$$
For the case $p=\infty$, I think if I can prove $|g(x)|\leq ||g||_{\infty}$ for almost every $x\in \mathbb{R}$, then I'll be done. Is this the most straightforward approach? Or is this intermediate result actually harder than trying the proof directly?
When $p< \infty$, I used the Fubini-Tonelli theorem to reduce the problem to showing that $$\int_{\mathbb{R}}|g|\mathrm{d}m \leq \int_{\mathbb{R}}|g|^p\mathrm{d}m$$ although again, this seems like it may be a roundabout way that is harder than a direct proof. Unfortunately, nothing else is jumping at me.
Best Answer
Assume WLOG $f,g$ are nonnegative. Claim:
$$ \int f(y)(g(x-y))^p\, dy < \infty$$
for a.e. x. Proof: If we integrate the left side with respect ot $x,$ we can switch the order of integration to get, by Tonelli, $(\int f)(\int g^p)< \infty.$ The claim follows.
So now write $g = g_1 +g_2,$ where $g_1= g\chi_{\{g\le 1\}}, g_2= g\chi_{\{g> 1\}}.$ We know $ \int f(y)g_1(x-y)\,dy < \infty$ for every $x.$ As for $g_2,$ note that $g_2 \le g_2^p.$ So
$$\int f(y)g_2(x-y)\,dy \le \int f(y)[g_2(x-y)]^p\,dy$$
and the latter is finite a.e. by the claim above.