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$X$ be a Polish space and $\mu, \nu$ be two any probability measure on it. Could anyone tell me how to construct a $\sigma$-finite probability measure $\lambda$ such that $\mu\ll \lambda$ and $\nu\ll \lambda$. Thanks!

Myshkin
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  • Probability measures are finite by definition (hence also $\sigma$-finite). So the condition on $\lambda$ is redundant. – drhab Sep 10 '19 at 09:58

2 Answers2

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Just take $\lambda =(\mu +\nu) /2$.

Kavi Rama Murthy
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    @GabrielRomon OP is talking about sigma finite probability measure! That sounds strange. I think he just means sigma finite measure. Anyway I have edited my answer. – Kavi Rama Murthy Sep 10 '19 at 09:57
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You could go for $\lambda=\frac12\mu+\frac12\nu$.

drhab
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