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Assume the set {∅} is a σ-algebra on ∅, related discussing is here.

So, is the tuple (∅,{∅}) a measurable space?

JJJohn
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    an algebra of sets must contain at least two elements – AlvinL Aug 07 '19 at 10:45
  • @AlvinLepik Thanks for your comments, please discuss that topic here – JJJohn Aug 07 '19 at 10:47
  • We have many such questions in mathematics. So it is important when to insert the word nonempty into a definition or statement. The oldest example is whether $1$ is a prime. Another example: is $\varnothing$ a topological space? If so, is it connected? Is it compact? – GEdgar Aug 07 '19 at 13:06

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According to this Book,

Definition 1.5 A measurable space (X, A) is a non-empty set X equipped with a σ-algebra A on X

your tuple is not a measurable space.

Jay
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  • Is there a particular reason for requiring that $X$ be nonempty? – Magma Aug 07 '19 at 10:55
  • If I want the category of measurable spaces and measurable maps to have the best properties of a category, then should I or should I not include this emptyset? – GEdgar Aug 07 '19 at 13:09
  • There is absolutely no reason for requiring that $X$ be nonempty except that some mathematicians have a pathological desire to exclude the empty set. See https://math.stackexchange.com/questions/45145/why-are-metric-spaces-non-empty for instance. – Eric Wofsey Aug 07 '19 at 23:08
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Yes, it is a measurable space. You may occasionally encounter some authors who require a measurable space to be nonempty but this is not the standard definition and it is a completely artificial and unnecessary restriction.

Eric Wofsey
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