Doubt in tibshirani elements of statistical learning book

Question

i am self studying the elements of statistical learning book, in chapter 2 there is this section link to book page here ]2

and i just dont get it. How does the author replace the integral with expectation and then conclude that it is sufficient to minimize the function point wise[link to second book page]1 also later he does the same for the categorical problem as well.

I dont understand these derivations and if someone could show how to do it and also if there are some general principles i should keep in mind when confronted by such terminology or derivations it would be really helpful as i am not a maths major and have never been very comfortable with these kind of things and have always been confounded by such leaps