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According to Wikipedia a semi-finite measure is defined as follows:

Definition: Let $( X ,\mathcal{ A} )$ be a measurable space and $\mu$ a measure on this measurable space. The measure $\mu$ is called an s-finite measure, if it can be written as a countable sum of finite measures $\nu_{n}(n \in N)$, $$ \mu = \sum\limits_{n=1}^{\infty}\nu_{n} $$

On the other hand there are several places where the following definition is used (Link)

Definition: Let $(X, \mathcal{M}, \mu)$ be a measure space. If for each $E \in \mathcal{M}$ with $\mu(E) = \infty$, there exists $F \in \mathcal{M}$ with $F \subset E$ and $0 < \mu(F) < \infty$, $\mu$ is called semifinite.

I think theses definitions are equivalent but how can I prove this?

bayes2021
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1 Answers1

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They are not equivalent. Let $\mu(A)=|A|$. Then $\mu$ satisfies the second definition, as every nonempty set contains a point, but not the first, because the real numbers are uncountable.

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