1. ## Convergence

Let $\displaystyle {g_n}$ be a sequence of nonnegative integrable functions defined on a measurable set $\displaystyle E$ such that $\displaystyle g_n \rightarrow g$ almost everywhere on $\displaystyle E$, where $\displaystyle g$ is integrable on $\displaystyle E$. Assume that $\displaystyle {f_n}$ is a sequence of measurable functions such that $\displaystyle |f_n| \leq g_n$ and $\displaystyle f_n \rightarrow f$ almost everywhere on $\displaystyle E$.
If $\displaystyle lim_{n \rightarrow \infty} \int_E g_n = \int_E g$, prove $\displaystyle lin_{n \rightarrow \infty \int f_n \rightarrow \int_E f}$.

Let $\displaystyle {g_n}$ be a sequence of nonnegative integrable functions defined on a measurable set $\displaystyle E$ such that $\displaystyle g_n \rightarrow g$ almost everywhere on $\displaystyle E$, where $\displaystyle g$ is integrable on $\displaystyle E$. Assume that $\displaystyle {f_n}$ is a sequence of measurable functions such that $\displaystyle |f_n| \leq g_n$ and $\displaystyle f_n \rightarrow f$ almost everywhere on $\displaystyle E$.
If $\displaystyle lim_{n \rightarrow \infty} \int_E g_n = \int_E g$, prove $\displaystyle lin_{n \rightarrow \infty \int f_n \rightarrow \int_E f}$.