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I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that

The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$.

Also, the distribution function induced by a probability $P$ on $({\mathbb R},{\mathcal B})$ is defined as $$ F(x)=P((-\infty,x]) $$

Is it because for some theoretical convenience that the half-open intervals are used often in probability theory or are they of special interest?

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One might just as well define $F(x) = P((-\infty,x))=1-P([x,\infty))$. One has to decide for one of these two options for consistency. While it does not matter in the end, it seems more naturally to have $F(6)=1$ for a die distribution function rather that $F(6)=\frac56$ and $F(6.000000001)=1$. –  Hagen von Eitzen Oct 14 '12 at 21:43
    
I think it depends on the textbook you're reading. The Borel set is a sigma algebra generated by open sets or equivalently half-open intervals. –  Pk.yd Oct 15 '12 at 11:57

3 Answers 3

up vote 2 down vote accepted

The fundamentally nice properties of half-open intervals are that:

  • They are closed under arbitrary intersections
  • For two half-open intervals $I_1, I_2$, their difference $I_1 \setminus I_2$ is a union of half-open intervals (a trivial union for $\Bbb R$, but not so in $\Bbb R^n$, in general)

That is, these half-open intervals form a so-called semiring of sets.

This is important because Carathéodory's theorem (on existence of measures) grips on such semirings; this route then leads to the theorem that Lebesgue measure on $\Bbb R^n$ exists.

I think this is one of the main reasons why probability and measure theorists like this type of interval.

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I think it's because the distribution function in the discrete case is the sum of probabilities from minus infinity up to and including x; but minus infinity is not a number so that end of the interval is open, i.e., has no end point.

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The half-open intervals are not necessarily special in a particular way, they are one of many possible generators of the Borel $\sigma$-algebra.

As I understand it, most of the things you do with half-open intervals you could also do with other generators, but in practice they are easy to work with

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