Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space, $(E, |\cdot|)$ a Banach space, and $p \in [1, \infty)$. In a previous thread, I proved that
Theorem: Let $f,g,f_n, g_n \in \mathcal L_p (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e., $g_n \to g$ $\mu$-a.e., $|f_n| \le |g_n|$ $\mu$-a.e., and $\|g_n\|_p \to \|g\|_p$. Then $\|f_n -f\|_p \to 0$.
Could you confirm if below related variant is indeed true? If not, please provide a counter-example.
Conjecture: Assume $f,g,f_n, g_n \in \mathcal L_1 (X, \mu, E)$ such that $f_n \to f$ $\mu$-a.e., $g_n \to g$ $\mu$-a.e., $|f_n| \le |g_n|$ $\mu$-a.e., and $\int g_n \to \int g$. Then $\int f_n \to \int f$.
