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In the first chapter of Probability wih Martingales (Willams) I came across the following relation

$$(-\infty,x]=\bigcap_{n\in N}(-\infty,\, x+n^{-1}).$$ Even though this is intuitive to me I would like to know if this is a rigorous statement. Is the intersection of a infinite number of sets defined in the same way as for a countable number of sets ? or by some limiting operation like liminf or limsup?

Thank you.

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To answer the title, the intersection of an infinite number of open sets is not necessarily open. –  Jean Hominal Aug 20 '13 at 8:15

2 Answers 2

up vote 6 down vote accepted

The intersection of any number of sets is defined in the following way:

$$\bigcap_{i\in I}A_i=\{a\mid\forall i\in I.a\in A_i\}.$$

In order to see that this equation is true, note that for every $y>x$ there is some $n$ such that $y-x>\frac1n$.

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thank you. I understand it. –  triomphe Aug 20 '13 at 9:35

The intersection is defined in the usual way: a real number $\alpha$ is in the intersection if and only if it belongs to each of the rays $(\leftarrow,x+n^{-1})$. That is the case precisely when $\alpha\le x$, so the intersection is indeed $(\leftarrow,x]$.

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thank you. nice explanation. –  triomphe Aug 20 '13 at 9:36
    
@MLT: You’re welcome. –  Brian M. Scott Aug 20 '13 at 16:57
    
Did you invent this arrow notation? I quite like it. –  Clive Newstead Aug 20 '13 at 21:53
    
@Clive: No, I picked it up somewhere; it was years ago, though, and I no longer have any idea where. –  Brian M. Scott Aug 21 '13 at 1:45

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