I'm reading a proof of a Khintchine inequality :
Let $(r_{1}, \dots , r_{n})$ be iid random variables with $P(r_{i} = \pm1) = \frac{1}{2}$. Let $f = \sum\limits_{j=1}^{n}a_{j}r_{j}$, where $a_{j} \in \mathbb{R}$.
Then $||f||_{2} \leq \sqrt{e}||f||_{1}$.
The proof uses Holder : $\forall g \in L^{\infty}, ||fg||_{1}\leq||f||_{1}||g||_{\infty}$, and then takes $$g = \prod\limits_{j=1}^{n}(1 + ia_{j}r_{j})$$
But I don't understand why we can take such a function, and still use Holder. I believe that $f,g$ must lie in $L^{p}$ spaces defined over the same space $S$ for that.