On one hand, according to wikipedia, it says that the pair $(X, \Sigma)$ is a measurable space, if $X$ is a set, and $\Sigma$ is a $\sigma$-algebra on $X$. To me, a "measurable space" means a space we can assign a meaningful measure to the measurable sets in the space.

On the other hand, it seems that the power set of any set is a $\sigma$-algebra on the set, so it means that if I take $X=\mathbb{R}$ and $\Sigma=2^{\mathbb{R}}$, then $(\mathbb{R}, 2^{\mathbb{R}})$ is a measurable space...but this is contrary to my current understanding (that for uncountable set $\mathbb{R}$, we usually need to find a smaller $\sigma$-algebra than $2^{\mathbb{R}}$, to define a meaningful measure).

What am I missing here?

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    $\begingroup$ What you seem to be missing is that words mean what the definitions say they mean - if the definition is not the same as what the word means "to you" then you need to forget about what it means "to you", because that's simply irrelevant. $\endgroup$ – David C. Ullrich Feb 23 '18 at 14:37
  • $\begingroup$ Thanks for your comments. You are right, I was vague in my previous understanding of the concepts related to $\sigma$-algebra (and measurable). Now, "measurable" is different from "usefully/meaningfully measurable", in the sense that "measurable" is an abstraction/generalization of properties of all kinds of measures (meaningful or not) which just satisfies the defintion of $\sigma$-algebra. $\endgroup$ – bruin Feb 24 '18 at 1:15
  • $\begingroup$ From this angle, the introduction of $\sigma$-algebra just like a definition of a base class in object-oriented languages, that the base class may not directly useful, but useful classes can be derived from the base class by adding some meaningful properties. E.g., "Borel" $\sigma$-algebra is a "useful" $\sigma$-algebra. $\endgroup$ – bruin Feb 24 '18 at 1:16

You are not missing anything: the word measurable space might just be a little misleading. Measurable space only means: a set $X$ and a $\sigma$-algebra $\Sigma$ on $X$. Nothing more. Nothing less.

  • $\begingroup$ Thanks for your answer. Strictly speaking, it seems that the word "measurable" is semantically not misleading, if I don't attach extra meanings (such as usefulness or nontriviality) to it. $\endgroup$ – bruin Feb 24 '18 at 1:34

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