I have some problems, help:
Let be a probability space. Let be a contraction such that is also a contraction. Suppose preserves positivity.
In a proof of why is a contraction for every , my lecture notes state that for all , we have
Why does the above equality hold? The strangest part to me is the fact that the LHS has a power of outside of the integral , but in the RHS the integral is not being raised to any power. I've tried to raise both sides to but I could not do much with this. If it helps, this came up in the study of conditional expectations.