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Using the weak law of large numbers

Hello,

To give a brief background

in my college math course on probability and statistics, the professor just finished going over Markov's inequality, Chebychev's inequality, and the Weak law of large numbers.

This is one of the examples given in the book, but I just can't seem to understand it. I especially do not understand the boxed step, and why the expected value of f(Ui) equals that integral.

If someone could provide a detiailed explanation, I would be very grateful.