I'm reading a proof which ends with:
$$ \ldots= \int_A \prod_{i\in D}p_\psi(y_i) d y_1 \cdots d y_n \leq \prod_{i\in D} \left[ \int p_\psi(y_i) d y_i \right] = 1 $$
I didn't follow the inequality where the product can be pulled outside the integral. Here $D$ is dataset, $\psi$ are the parameters of a probability distribution, $y_i$ is the $i$th iid observations of the unknown distribution in the dataset.
At first, I thought this might be an application of Cauchy-Schwarz but it seems like that introduces a squared term that I wouldn't know how to get rid of (https://proofwiki.org/wiki/Cauchy-Bunyakovsky-Schwarz_Inequality/Definite_Integrals).
Any explanation much appreciated! Thank you!