Measure Theory 8 | Monotone Convergence Theorem (Proof and Application) [dark version]

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Nice lesson!
but I have a doubt.
Isn't it inaccurate to say that the idea is that there is a f_n >= (1-epsilon)h almost everywhere as in the drawing?
if this were the case, uniform convergence would be needed (as in the case of the Reinman integral), right?

In this case it is enough that it is locally, i.e. on the X_n, which is what you use in the proof.
Or am I wrong?

tommasobellanova