It is known from work of Bennett and Gill [SIAM J. Comput., 10 (1981), pp. 96–113] and of Ambos-Spies [in Proc. 1st Structure in Complexity Theory Conference, 1986, pp. 23–34] that the following conditions are equivalent: (i) $L \in {\text{BPP}}$. (ii) For almost all oracles A, $L \in {\text{P}}^A $. It is shown here that the following conditions are also equivalent to (i) and (ii): (iii) The set of oracles A for which $L \in {\text{P}}^A $ has pspace-measure 1. (iv) For every pspace-random oracle A, $L \in {\text{P}}^A $. It follows from this characterization (and its proof) that almost every $A \in {\text{ESPACE}}$ is $ \leqslant _T^{\text{P}} $-hard for ${\text{BPP}}^A $. Succinctly, the main content of the proof is that pseudorandom generators exist relative to every pseudorandom oracle.
['Jack H. Lutz']
A pseudorandom oracle characterization of BPP