1. ## Convergence

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

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