L : { 0 , 1 } N → [0 , 1] is the limit average (5), and t is the test defined in Proposition 3 for the interval Q = [0 , 1]. From Corollary 1 and the definition of T it follows that for every realization x ∈ { 0 , 1 } N and every exchangeable distribution µ . We now claim that the desired properties of T follow from the corresponding properties of t . Indeed, for every exchangeable µ where the first equality follows from (7), the second from the definition of ¯ µ and the third from Proposition 3. Thus T accepts exchangeable data generating processes with probability 1. Let ζ ∈ Δ(Γ) and let ¯ ζ ∈ Δ (Δ([0 , 1])) be the push-forward of ζ under the map ν ↦→ ¯ ν where ¯ ν ∈ Δ([0 , 1]) is the pushforward of ν under L for every ν ∈ Γ. Fix a theory µ ∈ Γ and let ¯ µ be the pushforward of µ under L . Then where the first equality follows from (7) and the second from the definitions of ¯ ζ and ¯ µ . Therefore the set of all µ -s such that ζ ⊗ µ ( { ( ν, x ): T ( ν, x ) = PASS } ) = 0 is the preimage of the set of all ¯ µ such that ¯ ζ ⊗ ¯ µ ( { (¯ ν, q ): t (¯ ν, q ) = PASS } ) = 0 under the map µ ↦→ ¯ µ . Since the later set is co-meager by Theorem 3 and the map µ ↦→ ¯ µ is homeomorphism, it follows that the former set is co-meager, as desired. □ The expert testing literature asks whether an uninformed expert can produce forecasts that appear to be are as good as the forecasts of the informed given the observed outcomes. 5. Testing and merging 12 ζ ⊗ µ ({ ( ν, x ): x ∈ A ( T ) ν }) = ζ ⊗ µ ( { ( ν, x ): t (¯ ν, L ( x )) = PASS } ) = ¯ ζ ⊗ ¯ µ ( { (¯ ν, q ) | t (¯ ν, q ) = PASS } ) µ ( A ( T ) µ ) = µ ( { x | t (¯ µ, L ( x )) = PASS } ) = ¯ µ ( { q | t (¯ µ, q ) = PASS } ) = 1 (7) µ passes T over x ⇐⇒ t (¯ µ, L ( x )) = PASS