Levi's Monotone Convergence Theorems

# Levi's Monotone Convergence Theorems

We will now look at a group of theorems collectively known as Levi's Monotone convergence theorems. The first three of the theorems below regard increasing (almost everywhere on an interval $I$) sequences of step functions, upper functions, and Lebesgue integrable functions $(f_n(x))_{n=1}^{\infty}$ such that $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$.

Each of these three theorems have two important results. The first important result is that the sequence $(f_n(x))_{n=1}^{\infty}$ will converge to a function $f$. This is a very nice convergence result. If the limit of the integral of an increasing sequence of functions converges, then intuitively we should expect that the sequence of functions itself converges (otherwise we might as well expect that the numerical sequence of integrals would be divergent).

The second important result is that the limit of the integrals of the sequence functions will equal the integral of the limit of the sequence functions, i.e.:

(1)
\begin{align} \quad \lim_{n \to \infty} \int_I f_n(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \int_I f(x) \: dx \end{align}

We will not prove any of these theorems as they're rather cumbersome and technical.

 Theorem 1 (Levi's Theorem for Step Functions): If: 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of step functions, 2) $(f_n(x))_{n=1}^{\infty}$ is increasing almost everywhere on an interval $I$, 3) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists. Then: a) $(f_n(x))_{n=1}^{\infty}$ converges to some limit function $f \in U(I)$ almost everywhere on $I$. b) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \int_I f(x) \: dx}$.
 Theorem 2 (Levi's Theorem for Upper Functions): If: 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of upper functions, 2) $(f_n(x))_{n=1}^{\infty}$ is increasing almost everywhere on an interval $I$, 3) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists. Then: a) $(f_n(x))_{n=1}^{\infty}$ converges to some limit function $f \in U(I)$ almost everywhere on $I$. b) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \int_I f(x) \: dx}$.
 Theorem 3 (Levi's Theorem for Lebesgue Integrable Functions): If: 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions, 2) $(f_n(x))_{n=1}^{\infty}$ is increasing almost everywhere on an interval $I$, 3) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ exists. Then: a) $(f_n(x))_{n=1}^{\infty}$ converges to some limit function $f \in L(I)$ almost everywhere on $I$. b) $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \int_I f(x) \: dx}$.