How to prove that there always exist a simple random variable that approximates $X \in \mathcal{L}^p$?

Question

I have this problem on my Measure theory problem set:

We didn't prove this result in class directly, as a matter of fact. What we did prove, however, and I think is related sutff was that

1) There's always a sequence of simple random variables $(X_{n})$ that converges monotonically (pointwise) to $X$

2) Monotone Convergence Theorem

3) Dominated Convergence Theorem

I think 1) and 2) imply what the problem's text is saying "we did in class". Am I right? And how to proceed? Any ideas? Thanks a lot in advance!

Answer 1

I think 1) and 2) imply what the problem's text is saying "we did in class". Am I right?

This depends on which version of the monotone convergence theorem you are talking about. If you know only the most basic one (which is applicable for non-negative and increasing sequences), then this is no use to prove the assertion. However, since $X \in L^1$ is supposed to be integrable, we can apply the dominated convergence theorem instead:

For $X \in L^1$ denote by $X^+(\omega):= \max\{X(\omega),0\}$ and $X^-(\omega) := \max\{-X(\omega),0\}$ the positive and negative part, respectively. By statement 1), there exist sequences of non-negative simple random variables $(Y_n)_n$ and $(Z_n)_n$ which converge monotonically to $X^+$ and $X^-$, respectively. In particular, $0 \leq Y_n \leq X_n^+ \leq |X|$ and $0 \leq Z_n \leq X_n^- \leq |X|$. This implies that $$|Y_n-X^+| + |Z_n-X^-| \leq 4|X| \in L^1$$ and the left-hand side converges pointwise to $0$ as $n \to \infty$. Consequently, an application of the dominated convergence theorem gives for $$\int |(Y_n+Z_n)-X| \, d\mathbb{P} \xrightarrow[]{n \to \infty} 0.$$ If we choose $n \in \mathbb{N}$ sufficiently large, we find that the simple random variable $Y:=Y_n+Z_n$ satisfies $\|X-Y\|_1 \leq \epsilon$.

And how to proceed?

Actually, the reasoning is quite similar. Pick $X \in L^p$ and write $X = X^+ - X^-$. As in the first part of my answer, choose non-negative simple random variables $(Y_n)_{n \in \mathbb{N}}$ and $(Z_n)_{n \in \mathbb{N}}$ which converge monotonically to $X^+$ and $X^-$, respectively. Show that

$$|Y_n-X^+|^p + |Z_n-X^-|^p \leq 2^{p+1} |X|^p \in L^1$$

and conclude from the dominated convergence theorem that $$ \int |(Y_n+Z_n)-X|^p \, d\mathbb{P} \xrightarrow[]{n \to \infty} 0.$$