Real Analysis – Sigma Finite Measure and Absolute Continuity

measure-theoryreal-analysis

Let $\mu$ be a $\sigma$- finite measure on $(X,M)$. Prove that there exists a finite measure $\lambda$ on $M$ such that $\lambda\ll\mu$ and $\mu\ll\lambda$.

Can anyone give me a hint on how to start on this problem?

Best Answer

Some hints:

  1. What would happen if you have a measure $\lambda = f\,\mathrm d\mu$? Say, if $f > 0$ everywhere? Does integral of positive function w.r.t. $\lambda$ has to be infinite if $\lambda$ is infinite?

  2. To find such $f$, think of the fact that there exists an increasing sequence of sets on which $\lambda$ is finite, and union covers the whole space. How would you define $f$ on each of those sets?

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