Weighted Conditional Expectation in AdaBoost

I am looking at "Additive logistic regression a statistical view of boosting" paper (https://web.stanford.edu/~hastie/Papers/AdditiveLogisticRegression/alr.pdf)

In page 346, the authors introduce a definition for a weighted expectation. I do not understand the motivation behind this definition and how it is used to get to this result.

It starts introducing it to minimize the following expression:

Can someone explain the idea behind this please?

Thank you