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Step 1.Prove the property for $h$ which is an indicator function.
Step 2.Using linearity, extend the property to all simple positive functions.
Step 3. Using Monotone property extend the property to all $h∈mF^+$.
Step 4. Extend the property in question to $h∈L^1$ by writing $h=h^+−h^−$and using linearity.

This is a standard approach used to prove theorems in measure and probability theory (See Defn. 1.3.6 in http://statweb.stanford.edu/~adembo/nyu-2911/lnotes.pdf). What always confuses me is sometimes the theorem will be valid only for bounded measurable functions.

My question is - when can you extend the property to all $L^1$ functions, and when to only bounded measurable functions? What theorems are in play here? I believe the pi-lambda argument is also involved somehow, but I'm not sure how.

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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.

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