In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of measurable functions $f_1, f_2, \ldots, f_n$, then letting $f = f_1 f_2 \cdots f_n$, we have the inequality \begin{equation} \lVert f \rVert_r \leq \lVert f_1 \rVert_{p_1} \lVert f_2 \rVert_{p_2} \cdots \lVert f_n \rVert_{p_n}. \end{equation}

In particular, if $f_i \in L^{p_i}(X,\mu)$ for all $i$, then $f \in L^r(X,\mu)$.

I'm looking for a generalization of this inequality to infinite products. That is, suppose we have an infinite sequence $(p_i)_{i \in \mathbb{N}}$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^\infty \frac{1}{p_i} = \frac{1}{r}$ and an infinite sequence of measurable functions $(f_i)_{i \in \mathbb{N}}$. Suppose moreover that the function $f(x) = \lim_{n \to \infty} \prod_{i=1}^n f_i(x)$ exists for almost every $x$. Under what conditions can I assert that \begin{equation} \lVert f \rVert_r \leq \liminf_{n \to \infty} \prod_{i=1}^n \lVert f_i \rVert_{p_i} ? \end{equation}

It seems to me that this *might* follow automatically from the first version of Holder's theorem I quoted above, but I'm a little uncomfortable with taking the limits. Is there anything I should watch out for?

Thanks in advance!