Probability-1 by Albert N. Shiryaev

Author:Albert N. Shiryaev
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(52)

and since , we have

(53)

But, by the definition of ρ 1 ×ρ 2,

Hence it follows from (52) and (53) that (50) is valid for .

Now let . The set function

is evidently a σ-finite measure. It is also easily verified that the set function

is a σ-finite measure. As was shown above, λ and v coincide on sets of the form F = A × B, and therefore on the algebra . Hence it follows by Carathéodory’s theorem that λ and v coincide for all .

We turn now to the proof of the full conclusion of Fubini’s theorem. By (47),

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