for every n . Since ∏ n − 1 i =0 q s i (1 − q ) 1 − s i is a polynomial in q of degree n , it follows again by induction that for every n . Since a probability measure with bounded support is determined by its moments it follows that λ ′ = λ $^{′′}$. □ Corollary 1. There exists a Borel subset B of [0 , 1] N × { 0 , 1 } N and a Borel map f : B → Δ([0 , 1]) such that for every exchangeable theory µ and for every realization x = ( s$_{0}$, s$_{1}$, . . . ) one has where p$_{n}$ = µ (1 | s$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$ ) and ¯ µ is the push-forward of µ under the limit average L : { 0 , 1 } N → [0 , 1] defined in (5). Proof. Consider the Borel map g : Δ([0 , 1]) × { 0 , 1 } N → [0 , 1] N × { 0 , 1 } N that is given by g ( λ, s$_{0}$, s$_{1}$, . . . ) = ( p$_{0}$, p$_{1}$, . . . , s$_{0}$, s$_{1}$, . . . ) where p$_{n}$ = µ (1 | s$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$ ) for µ = ε$_{λ}$ . By the previous lemma this map is one-to-one. Therefore its image B = g ( Δ([0 , 1]) × { 0 , 1 } N ) is a Borel set and the inverse map g − 1 : B → Δ([0 , 1]) × { 0 , 1 } N is a Borel map (See Kechris (1994), Corollary 15.2). Take f = π ◦ g − 1 where π : Δ([0 , 1]) × { 0 , 1 } N → Δ([0 , 1]) is the projection on the first coordinate. The assertion in the corollary follows from the fact that if µ is exchangeable then µ = ε$_{λ}$ when λ = ¯ µ . □ Proof of Theorem 1. Consider the test T : [0 , 1] N × { 0 , 1 } N → { 0 , 1 } which is given by for every realization x = ( s$_{0}$, s$_{1}$, . . . ) ∈ { 0 , 1 } N and every sequence p = ( p$_{0}$, p$_{1}$, . . . ) ∈ [0 , 1] N of forecasts, where B ⊆ [0 , 1] N × { 0 , 1 } N and f : B → Δ([0 , 1]) are as in Corollary 1, 11 T ( p, x ) =     $^{t}$ ( f ( p, x ) , L ( x ) ) , if ( p, x ) ∈ B FAIL , otherwise. ∫ q $^{n}$λ $^{′}$(d q ) = ∫ q $^{n}$λ $^{′′}$(d q ) ( p$_{0}$, p$_{1}$, . . . , s$_{0}$, s$_{1}$, . . . ) ∈ B, and f ( p$_{0}$, p$_{1}$, . . . , s$_{0}$, s$_{1}$, . . . ) = ¯ µ.