# weird definition of supremum,inf of seq.? [closed]

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?
Thanks!

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## closed as not a real question by Andres Caicedo, Micah, kahen, ncmathsadist, Chris EagleFeb 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

## 1 Answer

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The approach of Halmos in doing everything on $\sigma$-rings instead of $\sigma$-algebras is a bit outdated, but the book is not bad.

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