This is exercise 11.2 in Folland's book:
Let $\mu$ be a Radon measure on the locally compact group $G$ and $f \in C_c(G)$. Prove that $$x \mapsto \int_G f(yx) \mu(dy)$$ is continous.
Before showing my attempt, let me quote a result that might be relevant:
If $f \in C_c(G)$, then $f$ is left and right uniformly continuous, i.e. for every $\epsilon > 0$, there is a neighborhood $V$ of $e$ (= the identity of $G$) such that $\sup_{x \in G} |f(y^{-1}x)-f(x)| \leq \epsilon$ and $\sup_{x \in G} |f(xy)-f(x)| \leq \epsilon$ for $y \in V$.
Attempt:
Note that $\operatorname{supp}(y \mapsto f(yx)) = \operatorname{supp}(f)x^{-1}$, so the integral makes sense since $\mu$ is finite on compact sets.
Let $x \in G$, $\epsilon > 0$. Choose a compact set $K$ outside which $f$ vanishes. Choose a compact neighborhood $K'$ of $x$. Applying the result above, there is a symmetric neighborhood $V$ of $e$ such that $$v \in V \implies \sup_{z \in G} |f(z)-f(zv)| \leq \frac{\epsilon}{\mu(K(K')^{-1})+1} \quad (*)$$ Note for this that $K(K')^{-1}$ is compact, hence its measure is finite.
Let $W:= K' \cap xV$, which is a neighborhood of $x$. Then if $x \in W$ $$\left|\int_G f(yx)\mu(dy) -\int_G f(yx') \mu(dy)\right| $$ $$\leq \int_G |f(yx)-f(yx')| \mu(dy)$$ $$= \int_{K(K')^{-1}}|f(yx)-f(yx')|\mu(dy)$$ $$= \int_{K(K')^{-1}}|f(yx)-f((yx)(x^{-1}x'))|\mu(dy)\leq \epsilon$$
The equality in the third line follows because $$yx \in K \implies y \in Kx^{-1}\subseteq K(K')^{-1}$$ so $$y \notin K(K')^{-1} \implies yx \notin K \implies f(yx) = 0$$
Similarly, $yx' \in K \implies y \in K(x')^{-1} \subseteq K(K')^{-1}$ gives us that $f(yx') = 0$ if $y \notin K(K')^{-1}$
The last inequality follows because $x^{-1}x' \in x^{-1}xV = V$ so we can apply $(*)$. This shows that we have continuity at $x$ and thus ends the proof. $\quad \square$
Questions:
(1) Is this proof correct?
(2) Would it be correct that the statement is more generally true for Borel measures that are finite on compact sets (as my proof seems to suggest?)
(1): Almost, the only error is that you need to choose $x' \in W$ before you estimate the difference of the integrals (this is probably a typo).
(2): Yes, any Borel measure that is finite on compact sets suffices here.