Mathematics for Econometrics by Phoebus J. Dhrymes

Author:Phoebus J. Dhrymes
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

ii.If A is of σ-finite measure, but μ(B) = 0 does not hold, then we can only assert that

where the sup is taken over all bounded measurable functions, g, such that g ≤ f and g vanishes outside a set, say A, of finite measure.

The proof of this Proposition is beyond the scope of this volume.

Remark 7.22.

The preceding discussion concludes the demonstration that the results obtained in Propositions 7.11 through 7.19, remain valid even when dealing with abstract measure spaces.

We now turn to the discussion of another topic, which will play an important role when we consider conditional probability and conditional expectation. We have

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