We will review the generalized Neyman-Pearson lemma, see [CK01].

Let be a probability space. are two subspaces of -integrable random variables. Let be defined by

and

The problem is to find

In other words, it is to maximize

over the constraints

**Assumption 1** * is convex and closed under -a.e. *

The following theorem is a straightforward generalization of the result of [CK01].

**Theorem 1** * Suppose Assumption 1holds. Then, there exists , such that*

- For all , following holds:
This implies the exitence of the saddle point, i.e.

- Let . Then,
Moreover, is given by

- For all , there exists such that
can be taken by and .

- can take the form of
where is chosen to satisfy

- In such a way, we have
and .

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