When is a measurable function a Bayesian decision function? How do I prove this?
Can you give an example with standard or weighted binary classification?
This question is very easy to answer. It targets to the area of machine learning which is teached in university courses. A Measurable function is simply the policy of an agent, a bayesian decision function is used in area of reinforcement learning for determine the actions of an agent, and the term “standard binary classification” separates the measured data in two groups with a hyperplane. What is more difficult to answer is the question, how the terms can be used in a concrete problem, for example to navigate a robot through a maze. But this wasn't asked by the OP.
The questions is a bit strange. I would say that the term Bayesian decision function definitely requires explanation. It is not widely used and does not have a commonly known definition.
But given that you mentioned a binary classification, I would assume that bayesian decision function this is a map from a countable set (observed outcomes) to another such set (actions). I also think that the pre-image of the total union of all actions should cover the whole set of possible outcomes. And as long that that last property holds and that function is also measurable than it would be a the bayesian decision function.