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The following is a cross-post of this question on math.SE, which did not attract any comment and may therefore be too research-oriented for math.SE.

It is a common technique in measure theory to prove something for indicator functions / elementary functions, generalize it to positive-valued functions and to measurable functions via $X=X^+ − X^−$. It is used e. g. to prove multiple integral properties or "independent random variables are uncorrelated".

This technique is a bit boring to execute, if one has shown a question already for the indicator functions. Is it somehow possible to describe the set of all statements which are true for all measurable functions if the result is proven for only indicator functions / elementary functions? I'm in search of something like the Transfer Principle in nonstandard analysis.

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I think, you are looking for a "monotone class argument", see the wikipedia entry for the monotone class theorem. In a nutshell, prove that the property holds for indicators and is preserved under finite sums, scalar multiplication and under monotonically increasing limits and conclude that it holds for the full class.

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