how to prove that every positive r.v. X is the non-decreasing limit of a sequence of simple r.v.’s

Question

Every positive r.v. X is the non-decreasing limit of a sequence of simple r.v.’s. That is, if $X \geq 0$, then there exist simple r.v.’s Xn, defined on the same probability triple, such that:
$\mathbf{x_n(\omega) \uparrow X(\omega) \quad \forall \quad \omega \in \Omega}$

The above proposition can be proved by constructing intervals

$\mathbf{A_jn=\{ \frac{j}{2^n} < X \leq \frac{j+1}{2^n} \} \quad for \quad j=0,1,2,...}$

Could anybody explain the steps? I cannot grasp the concept.

Answer 1

Here is one standard construction:

Set $x_n(\omega) := 2^{-n} \lfloor (X(\omega)\wedge n) 2^n \rfloor$.

• Clearly, every $x_n$ is smaller than $X$ for each $n$ at each point $\omega$ (we are rounding down, not up).
• For each $x\in\mathbb{R}$, the set $x_n^{-1}([x,\infty))$ is a finite union of sets $A_{jn}$. Each set $A_{jn}$ is measurable (because $[j/2^n, (j+1)/2^n]$ are Borel sets, and $X$ is a random variable, and therefore measurable). Hence, every $x_n$ is measurable.
• For each $n$, you can represent $x_n$ as a finite sum of the form $\sum_{j}1_{A_{jn}}c_{jn}$, because we have truncated $X$ by $X\wedge n$ before approximating it from below.

For the intuition: Imagine the area under the graph of a function $X: \mathbb{R} \to [0,\infty)$. At step $n$, you cut off everything above the level $n$, and then try to fill the space below the curve with vertical bars that have heights equal to multiples of $2^{-n}$. For example, in the following image, you see a curve that represents a measurable function $X$. In the first step, you are trying to fill everything below $1$ by chunks of height $1/2$ (big green chunks). In the second step, you are filling everything under $2$ by chunks with height $1/4$ (finer yellowish chunks). Then you fill everything up to the level $3$ with chunks of height $1/8$ (finer dark-red-brown part). In the forth step, you fill the area under $X \wedge 4$ with fine dark-red pieces. You proceed with bright red pieces to fill the tiny remaining gaps etc. The graphs that delimit the colored areas that you filled in $n$-th step are the $x_n$.