Lebesgue's Monotone Convergence Theorem for Series

Theorem 1 (Lebesgue's Monotone Convergence Theorem for Series): 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) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ converges on $E$. Then $\displaystyle{\sum_{n=1}^{\infty} \int_E f_n = \int_E \sum_{n=1}^{\infty} f_n}$.

• Proof: Let $(f_n(x))_{n=1}^{\infty}$ be a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$. For each $k \in \mathbb{N}$ define a new function $F_k$ as:

(1)

\begin{align} \quad F_k(x) = \sum_{n=1}^{k} f_n(x) \end{align}

• Then each $F_k$ is a finite sum of nonnegative Lebesgue measurable functions. So each [$F_k$ is a nonnegative Lebesgue measurable function. Furthermore, $(F_k(x))_{k=1}^{\infty}$ is a sequence of nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$ such that $0 \leq F_1(x) \leq F_2(x) \leq ... \leq F_n(x) \leq ...$ and $(F_k(x))_{k=1}^{\infty}$ converges pointwise to $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ on $E$ from (1). By the Lebesgue's Monotone Convergence Theorem we have that:

(2)

\begin{align} \quad \lim_{n \to \infty} \int_E F_k = \int_E \sum_{n=1}^{\infty} f_n \quad (*) \end{align}

• By the linearity of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:

(3)

\begin{align} \quad \lim_{n \to \infty} \int_E F_k &= \lim_{n \to \infty} \int_E (f_1 + f_2 + ... + f_k) \\ &= \lim_{n \to \infty} \sum_{n=1}^{k} \int_E f_n \\ &= \sum_{n=1}^{\infty} \int_E f_n \quad (**) \end{align}

• From $(*)$ and $(**)$ we have that:

(4)

\begin{align} \quad \sum_{n=1}^{\infty} \int_E f_n = \int_E \sum_{n=1}^{\infty} f_n \quad \blacksquare \end{align}