Let $X$ be a measure space. Let $S_j$, $j \in \mathbb N$ be an increasing sequence of $\sigma$-algebras on $X$ such that $S := \bigcup_{j \geq 0} S_j$ is a $\sigma$-algebra. For every $j$, let $\mu_j$ be a probability measure on $S_j$.

Let $f_{ij}$ $(i, j \in \mathbb N)$ be a double indexed sequence of functions such that that for every $j$, $f_{ij}$ converges $\mu_j$-a.e. to a $S_j$ measurable function $f_j$.

Suppose there exists some probability measure $\mu$ on $S$ such that $f_j$ converges $\mu$-a.e. to a function $f$.

Suppose further that:

the restriction of $\mu$ to $S_j$ is absolutely continuous with respect to $\mu_j$ for every $j$

$f_{ij}, f_j, f$ are $\mu$-integrable, and $f_{ij}$ is $\mu_j$-integrable for every $i, j$

$\int f_{ij} \, d \mu_j \to \int f_j \, d \mu_j$ for every $j$

$\int f_j \, d \mu \to \int f \, d \mu$

Is it true that there exists a increasing function $b: \mathbb N \to \mathbb N$ such that $\int f_{n, b(n)} \, d \mu$ converges to $\int f \, d \mu$?