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in Paul R. Halmos - Measure Theory: for a sequence ${x_n}$ the supremum of the sequence is \, and the infimum of $x_n$ is the intersection
of $x_n$ for n from 1 to infinity, how to understand that?
by the way is it a right book for undergrad study for measure theory?

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closed as not a real question by Andrés E. Caicedo, Micah, kahen, ncmathsadist, Chris Eagle Feb 27 '13 at 2:05

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I don't understand "the supremum of the sequence is \,". – Gerry Myerson Feb 26 '13 at 11:34
up vote 2 down vote accepted

If $(B_n)$ is a sequence of subsets of $X$, then the pointwise supremum of the indicator functions $(1_{B_n})$ coincides with the indicator function of $\bigcup_n B_n$, so it makes sense to call $\bigcup_n B_n$ the supremum of the sequence $(B_n)$. It is also the order theoretic supremum under the relation $\subseteq$ on the powerset of $X$.

The approach of Halmos in doing everything on $\sigma$-rings instead of $\sigma$-algebras is a bit outdated, but the book is not bad.

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