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===Proof===
The following argument originates in [[Rick Durrett]]'s Probability: Theory and Examples.
<ref name="Durrett">{{cite book|last=Durrett|first=Rick|year=2010|title=Probability: Theory and Examples|edition=4th|publisher=Cambridge University Press|page=
The assumption <math>\Omega\, \in \mathcal{A}</math>, (2) and (3) imply that <math>\mathcal{G} = \{A: \mathbf{1}_{A} \in \mathcal{H}\}</math> is a ''λ''-system. By (1) and the [[Dynkin system|{{pi}}−''λ'' theorem]], <math>\sigma(\mathcal{A}) \subset \mathcal{G}</math>. (2) implies <math>\mathcal{H}</math> contains all simple functions, and then (3) implies that <math>\mathcal{H}</math> contains all bounded functions measurable with respect to <math>\sigma(\mathcal{A})</math>.
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