Using Lebesgue's Dominated Convergence Theorem to show convergence of a sequence of random variables

Question

Let $ \xi$ and $\eta$ be random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}[|\xi|], \mathbb{E}[|\eta|] < + \infty$.

I'd like to show that $\xi$ can be approximated by the sequence of random variables $\{\xi_n\}$ (defined below) in the sense that

$$ \lim_{n \rightarrow \infty} \int_\Omega \xi_n \text{ d}\mu = \int_\Omega \xi \text{ d}\mu $$

where $$ \xi_n = \sum_{k=1}^n b_k \mathbb{1}_{B_k}$$

where $ n\in \mathbb{N}$, $B_k \in \mathcal{F}$, $b_k \in \mathbb{R}$ $\forall k$, and $\mathbb{1}_{B_k}$ are the corresponding indicator functions.

By Lebesgue's dominated convergence theorem, it suffices to show that $\{\xi_n\}$ converges to $\xi$ pointwise and is dominated by some integrable function $g$ in the sense that $\forall n \in \mathbb{N}$, $\forall x\in \Omega$, $$ |\xi_n(x)| \leq g(x)$$

If this is true, then $\eta$ can also be approximated by a sequence of random variables $\{\eta_n\}$ in the above sense, with

$$ \eta_n = \sum_{k=1}^n a_k \mathbb{1}_{A_k}$$

for some $A_k \in \mathcal{F}$, $a_k \in \mathbb{R}$

I'd also like to show that

$$ \lim_{n \rightarrow \infty} \int_\Omega \xi_n\eta_n \text{ d}\mu = \int_\Omega \xi \eta \text{ d}\mu $$

Here my idea is to write

$$|\xi_n \eta_n - \xi\eta| = |\xi_n \eta_n - \xi\eta_n + \xi\eta_n - \xi\eta|$$ $$ \leq |\eta_n(\xi_n - \xi)| + |\xi(\eta_n - \eta)|$$ by the triangle inequality. If $|\xi_n - \xi| \rightarrow 0$, $|\eta_n - \eta| \rightarrow 0$, then $|\xi_n \eta_n - \xi\eta| \rightarrow 0$.

However, I'm having trouble showing the remaining assumptions in Lebesgue's dominated convergence theorem, namely, that:

• $\{\xi_n\}$ converges to $\xi$ pointwise
• $\{\xi_n\}$ is dominated by some integrable function

It is also not clear how to show that $\{\xi_n\eta_n\}$ is dominated by some integrable function, since the product of two Lebesgue-integrable functions (the ones dominating $\{\xi_n\}$ and $\{\eta_n\}$ isn't necessarily Lebesgue-integrable.

Any help/advice would be greatly appreciated!

Answer 1

Using @FShrike's hint:

First assume $\xi: \Omega \rightarrow \mathbb{\bar{R}}$ is a non-negative function: $$\xi \geq 0$$

By the Simple Approximation Theorem, there exists a sequence of real valued simple functions $\{\xi_n\}_{n=1}^{\infty}$ on $\Omega$ such that $$ 0 \leq \xi_1 \leq \xi_2 \leq ... \leq \xi$$ and $\xi_n \rightarrow \xi$ pointwise.

This sequence $\{\xi_n\}_{n=1}^{\infty}$ satisfies all the conditions of Lebesgue Dominated Convergence Theorem (LDCT), from which we deduce that

$$\xi_n \rightarrow \xi \text{ in the } L_1 \text{ norm.}$$

In the general case when $\xi: \Omega \rightarrow \mathbb{\bar{R}}$ is not necessarily non-negative on all of $\Omega$, we can write $\xi$ as $$ \xi = \xi^{+} - \xi^{-}$$ where $\xi^{+} = \max(\xi, 0)$, $\xi^{-} = \max(-\xi,0)$.

Then $\xi_n^{+} \rightarrow \xi^{+}$ and $\xi_n^{-} \rightarrow \xi^{-}$ in the $L_1$ norm, whence there is a sequence of simple functions $\{\xi_n^{+} - \xi_n^{-}\}_{n=1}^{\infty}$ that converges to $\xi$ in the $L_1$ norm. $\square$

Exactly the same logic can be applied to show that $\xi \eta$ can be approximated by a sequence of simple functions (the product of two measurable functions is measurable, so we can once again apply the Simple Approximation Theorem followed by LDCT).