Lebesgue's Monotone Convergence Theorem

Theorem 1 (Lebesgue's Monotone Convergence Theorem): Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$. Suppose that: 1) $0 \leq f_1(x) \leq f_2(x) \leq ... \leq f_n(x) \leq ... \leq f(x)$ for all $x \in E$. 2) $(f_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$. Then $\displaystyle{\lim_{n \to \infty} \int_E f_n = \int_E f}$.

• Proof: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$ which satisfies (1) and (2) above. From (1) and the monotonicity property of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:

(1)

\begin{align} \quad 0\leq \int_E f_1 \leq \int_E f_2 \leq ...\leq \int_E f_n \leq ... \leq \int_E f \end{align}

• Therefore $\displaystyle{\left ( \int_E f_n \right )_{n=1}^{\infty}}$ is an increasing sequence of numbers that is bounded above by $\displaystyle{\int_E f}$. Therefore:

(2)

\begin{align} \quad \lim_{n \to \infty} \int_E f_n \leq \int_E f \quad (*) \end{align}

• Furthermore, from (2), since $(f_n(x))_{n=1}^{\infty}$ is a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$ that converges pointwise to $f(x)$, from Fatou's Lemma for Nonnegative Lebesgue Measurable Functions we have that:

(3)

\begin{align} \quad \int_E f \leq \liminf_{n \to \infty} \int_E f_n \leq \lim_{n \to \infty} \int_E f_n \quad (**) \end{align}

• From $(*)$ and $(**)$ we conclude that:

(4)

\begin{align} \quad \lim_{n \to \infty} \int_E f_n = \int_E f \end{align}