From what I understood
a Measurable Space is $(X,S)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ algebra
a Measure Space is $(X,S,\mu)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ algebra and $\mu$ is a measure.
Is it the same like a metric space? we can a general space and then add a metric so it becomes a metric space?
Does all Measurable Space can be a Measure Space? (as the name suggests)
$\endgroup$1 Answer
$\begingroup$You can always define a measure on any sigma algebra. Take any point $x \in X$ and define $\mu (E)=1$ if $x \in E$ and $0$ otherwise. So any measurable space can be made into a measure space.
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