Suppose that are i.i.d. to F, an unknown distribution. We wish to estimate the true mean of F with the loss function

Prove that is consistent for ERM induction.

Proof so far.

Now, I know that the Bayes learner and the ERM learner

I need to show that both the risk function and the empirical risk function converges to the Bayes Risk

That is, I need to show that and

For the first part, I have:

Looks like I'm lost...

I know by the the Law of Large Number, I have , so I'm basically trying to mold it but stuck...

I'm guessing that one mistake I made is that the loss function should be , but if I try that way, I can't get and I still can't get the ERM consistency going...

Any help, please? Thank you!!!