On a probability space \( (\Omega, \mathcal{B}, P) \), let \( \{f_n \), \( n=1 \), \( 2,\dots\} \) be a sequence of nonnegative random variables in \( L_1 \) such that \( f_n \to f\in L_1 \) with probability 1 and define \( g=\sup_n f_n \). If \( g\in L_1 \), the Lebesgue dominated convergence theorem asserts that \( E(f_n)\to E(f) \). More generally, as noted by [Doob 1953, p. 23], if \( g\in L_1 \), then for any Borel field \( \mathcal{B}_0 \) contained in \( \mathcal{B} \),
\[ E(f_n\mid\mathcal{B}_0) \to E(f\mid\mathcal{B}_0) \quad\text{a.e.} \]
If one extends this result in a minor manner, Lebesgue’s condition \( g\in L_1 \) is not only sufficient but necessary, as the following converse to the dominated convergence theorem asserts.
If \( f_n \geq 0 \), \( f_n \to f \) a.e., \( f_n\in L_1 \), and \( g = \sup_n f_n \notin L_1 \), there are, on a suitable probability space, random variables \( \{f_n^* \), \( n=1 \), \( 2,\dots\} \), \( f^* \), and a Borel field \( \mathcal{C} \) such that \( f^* \), \( f_1^* \), \( f_2^*,\dots \) have the same joint distribution as \( f \), \( f_1 \), \( f_2,\dots \), and
\[ P\bigl\{E(f_n^*\mid\mathcal{C})\to E(f^*\mid\mathcal{C})\bigr\} = 0. \]
