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.