I am reading a text on convex analysis and the following result appears in it
$ \textbf{Proposition:}$
Let $f_1,f_2$ be nonnegative functions, let $p>0$ then we define: $\begin{matrix}\\ f_1\ast _p f_2: & \mathbb{R}\rightarrow \mathbb{R} & \\ & x \mapsto & \left ( \int _\mathbb{R}\left [ f_1(y)f_2(x-y) \right ]^{p}dy \right )^{1/p} \end{matrix}$
called convolution of order $p$, then this integral converges to $\sup_{y\in\mathbb{R}} \left \{ f_1(y)f_2(x-y) \right \}$ when $p\rightarrow \infty$
Proof:
This is a problem of the convergence of $\left \| . \right \|_p$ to $\left \| . \right \|_\infty$ as follows:
Define $\forall x\in \mathbb{R}: f_x(y):=f_1(y)f_2(x-y)$ then
$\left \| f_x \right \|_p= \left ( \int _\mathbb{R}\left | f_x(y) \right |^{p}dy \right )^{1/p} =\left ( \int _\mathbb{R}\left | f_1(y)f_2(x-y) \right |^{p}dy \right )^{1/p}=\left ( \int _\mathbb{R}\left [ f_1(y)f_2(x-y) \right ]^{p}dy \right )^{1/p}=f_1\ast_p f_2(x)$
Analogously, it is the case that
$\left \| f_x \right \|_\infty=\sup_{y\in\mathbb{R}} \left \{ f_1(y)f_2(x-y) \right \}$
and we know that the p-norm converges to the infinite norm when p tends to infinity
My question is related to the domain of the functions, since we know that the above is true if $f_x$ is defined in a compact domain and if it is continuous.
Now in the text there is no mention of more properties except that the functions are nonnegative.
For this result to be fulfilled I should assume that $f_x$ is continuous and defined in a compact domain, or in fact that function can be defined in all the real line?