Dominated convergence theorem

Theorem (Lebesgue) Let $f_n$ be a sequence of measurable functions in the measure space $(X,\Sigma ,\mu )$ such that $f_n\to f$ (pointwise convergence) a.e. Then if there exists a measurable function $g$ such that $|f_n|\le g$ a.e. for all $n$, and $g$ is integrable,

$$\underset{n\to \infty }{\lim }\int |f-f_n|d\mu =0$$

that is $f_n\stackrel{L^1}{\to }f$. Remark Not that $L^1$ convergence also implies

$$\underset{n\to \infty }{\lim }\int f_{n}d\mu =\int fd\mu$$

i.e. we can exchange the limit with the integral. This follows from this simple inequality:

$$\left|\int fd\mu -\int f_{n}d\mu \right|\le \int |f-f_n|d\mu$$

Proof We will use the reverse Fatou’s lemma on the positive functions $|f-f_n|$ which are bounded by $2g$:

$$\underset{n}{\mathrm{limsup}}\int |f-f_n|d\mu \le \int \underset{n}{\mathrm{limsup}}|f-f_n|d\mu =0\square$$