Standard machine in measure theory

You're getting confused because you are getting the "standard machine" mixed up with the monotone class theorem. The standard machine is used when you want to show some property holds for all $L^1$ functions by proving them for indicators and invoking the monotone convergence theorem. The thing is that (at least on the surface), you have to prove the property in question for indicators of general measurable sets.

On the other hand, the monotone class theorem is used to prove properties for general sets of functions (not necessarily the set of all $L^1$ functions) by first showing them for indicators of sets in a generating $\pi$-class. The trade-off here is that the set of functions you are dealing with has to be a monotone class (of functions).

Traditionally, this theorem is stated only for vectors spaces of bounded measurable functions, but as Dembo notes in your link, it's not hard to extend it to general measurable functions.

Why are logarithms not defined for 0 and negatives?

$T$ surjective iff $T^*$ injective in infinite-dimensional Hilbert space?

What is the precise definition of a rigid shape?

Perfect circles in the Mandelbrot set?

Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.

Is there a simple proof for ${\small 2}\frac{n}{3}$ is not an integer when $\frac{n}{3}$ is not an integer?

Is the symmetric definition of the derivative equivalent?

Smooth sawtooth wave $y(x)=\cos(x-\cos(x-\cos(x-\dots)))$

If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$

Proof that a subset closed under group operation of a finite group is a subgroup

Value of $\lim_{n\to\infty}{(1+\frac{2n^2+\cos{n}}{n^3+n})^n}$

Does this intuition for "calculus-ish" continuity generalize to topological continuity?