Question

In the measurable space (R, B), the two measures ν = λ and µ = ∞ · λ are given. So, µ(A) = 0 if λ(A) = 0, ∞ otherwise. Show that ν ≪ µ but ν cannot be written as an integral with respect to µ.

Name: in the measurable space r b the two measures and are given so a 0 if a 0 otherwise show that but cannot be written as an integral with respect to 71883
Uploaded: 2022-11-28T11:59:23-08:00
Duration: 6 min 40 s
Channel: Sri K
Description: in the measurable space r b the two measures and are given so a 0 if a 0 otherwise show that but cannot be written as an integral with respect to 71883

          In the measurable space (R, B), the two measures ν = λ and µ = ∞ · λ are given. So, µ(A) = 0 if λ(A) = 0, ∞ otherwise. Show that ν ≪ µ but ν cannot be written as an integral with respect to µ.

Added by Nathaniel T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

11/28/2022

Step 1

This is because if µ(A) = 0, then A is a null set with respect to µ, and by the definition of absolute continuity, ν ≪ µ if for every null set A with respect to µ, ν(A) = 0. Show more…

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In the measurable space (R, B), the two measures ν = λ and µ = ∞ · λ are given. So, µ(A) = 0 if λ(A) = 0, ∞ otherwise. Show that ν ≪ µ but ν cannot be written as an integral with respect to µ.