Definition 10. *A hypothesis space family* H = fHg is permissible if there exist sets S and T *that* are analytic subsets of Polish spaces S and T *respectively, and a function* f : Z - T - S ! [0; 1℄,
measurable with respect to S  B(T )  B(S)*, such that*

```
                        H =
                           
                             ff (  ; t; s) : t 2 T g : s 2 S
                                                 
                                                   :
                                                   
  Let (X; ; ) be a measure space and T be an analytic subset of a Polish space. Let A(X)
denote the analytic subsets of X. The following three facts about analytic sets are taken from
Pollard (1984), appendix C.

```

(a) If (X; ; ) is complete then A(X)  .

(b) A(X - T ) contains the product -algebra   B(T ).

(c) For any set Y in A(X - T ), the projection XY of Y onto X is in A(X).

Recall Definition 2 for the definition of H . In the following Lemma we assume that (Z; B(Z))
has been completed with respect to any probability measure P , and also that (P; B(P)) is complete with respect to the environmental measure Q. Lemma 32. *For any permissible hypothesis space family* H ,

```
1. H n
     l
      
       is permissible.

```

2. fh 2 H: H 2 H g *is permissible.* 3. H is permissible for all H 2 H .

4. supH and infH are measurable for all H 2 H .

5. H*is measurable for all* H 2 H .
6. H is permissible.

Proof. As we have absorbed the loss function into the hypotheses h, H n l is simply the set of all n-fold products H      H such that H 2 H . Thus (1) follows from Lemma 31. (2) and (3)
are immediate from the definitions. As H is permissible for all H 2 H , (4) can be proved by an identical argument to that used in the "Measurable Suprema" section of Pollard (1984), appendix C.

For (5), note that for any Borel-measurable h : Z ! [0; 1℄, the function h : P ! [0; 1℄ defined by h(P ) := R
Z h(z) dP (z) is Borel measurable Kechris (1995, chapter 17). Now, permissibility of H automatically implies permissibility of H := fh : h 2 Hg, and H = infH so His measurable by (4).

Now let H be indexed by f : Z - T - S ! [0; 1℄ in the appropriate way. To prove (6),
define g : P - T - S ! [0; 1℄ by g(P; t; s) := R
Z f (z; t; s) dP (z). By Fubini's theorem g is a B(P) 
 B(T ) 
 B(S)-measurable function. Let G: P - S ! [0; 1℄ be defined by G(P; s) :=
inf t2T g(P; t; s). G indexes H in the appropriate way for H to be permissible, provided it can be shown that G is B(P)  B(S)-measurable. This is where analyticity becomes important. Let g := f(P; t; s) : g(P; t; s) > g. By property (b) of analytic sets, A (P - T - S) contains g.

The set G := f(P; s) : G(P; s) > g is the projection of g onto P - S, which by property (c) is also analytic. As (P; B(P); Q) is assumed complete, G is measurable, by property (a). Thus G is a measurable function and the permissibility of H follows.