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a) $(a,b)$

b) Any finite set.

c) The set of natural numbers.

d) The set of $\mathbb R-\mathbb Q$=irrational numbers.

I will appreciate your answers..

  • 0
    Good: you have at least seen the definition of a $\sigma$-algebra. One last question (and something I should have asked earlier): when your instructor said that "the Borel $\sigma$-algebra on $\mathbb{R}$ is generated by the class of intervals," was s/he referring only to _open_ intervals, or were all kinds of intervals allowed to be in the generating set (such as $( 0 , 1]$ or $[-1,\pi]$)? (The only difference this makes is in the details.)2012-10-29

1 Answers 1

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First, assume your teacher by interval meant any of the following kinds of intervals: $(a,b)$, $\ [a,b]$, $\ [a,b)$ and $(a,b]$.

(As an exercise, you should try to prove that each can be obtained by any fix kind of intervals, using countably infinite union and/or intersection of intervals of the fixed kind.)

  • Then a) is immediate.
  • For b), try to express the one point set $\{a\}$ as a countable intersection of intervals.
  • Then use countable union of these one point sets for b) and c), and also for d), showing that $\Bbb Q$ is also Borel (for this you should know what the cardinality of $\Bbb Q$ is.)
  • Then, for d) use the $\sigma$-algebra requirement: closedness under set complement.
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    @idobi182: You can, and should, [accept this answer](http://meta.math.stackexchange.com/questions/3286/how-do-i-accept-an-answer) if you found it useful and helpful.2012-11-01