A Concise Introduction to Measure Theory by Satish Shirali

Author:Satish Shirali
Language: eng
Format: epub, pdf
ISBN: 9783030032418
Publisher: Springer International Publishing

7.1.6. Notation.

Given a function g: [a, b] → , its extension to all of obtained by setting it equal to 0 outside [a, b] will be denoted by g*.

It is clear that and . Recall from Sect. 4.​4 that the measure induced on [a, b] by Lebesgue measure on is nothing but Lebesgue measure on [a, b]. It follows upon using Proposition 4.​4.​1 that is measurable if and only if is. Moreover, when this is so, g is integrable if and only if is, in which case .

It is trivial to see that .

7.1.7. Corollary.

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