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LeonhardEuler
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Probably a stupid question here, but I've been beating myself up over it and can't find a resolution. I'm reading a book (Probability: Theory and Examples by Rick Durrett) that defines a semialgebra as:
Already, this seems extremely odd to me because the compliment of a set belongs to the same space as the set itself. The disjoint union introduces another index to each element, if I am understanding that correctly as according to http://mathworld.wolfram.com/DisjointUnion.html" . So unless we are dealing with strange sets that include elements of different dimensions, I don't see how this is possible. The book then goes on to show that I clearly have misunderstood something because it then gives an example of a semialgebra:
But if i look at the interval (0,1] in R, then its compliment is
[tex](-\infty,0] \cup (1,\infty)[/tex]
Which is a union of intervals of the real line, not a disjoint union. A disjoint union would seem to have sets of the form [tex](-\infty,0] \times {{0}}[/tex] which don't belong to the real line at all.
Where am I going wrong?
Durrett said:A collection of sets S is said to be a semialgebra if (i) it is closed under intersection, and (ii) if S is an element of S, then Sc is a finite disjoint union of sets in S.
Already, this seems extremely odd to me because the compliment of a set belongs to the same space as the set itself. The disjoint union introduces another index to each element, if I am understanding that correctly as according to http://mathworld.wolfram.com/DisjointUnion.html" . So unless we are dealing with strange sets that include elements of different dimensions, I don't see how this is possible. The book then goes on to show that I clearly have misunderstood something because it then gives an example of a semialgebra:
Durrett said:An important example of a semialgebra is Rdo = the collection of sets of the form
(a1,b1]X ... X(ad,bd] , a subset of Rd where [tex]-\infty \leq a_{i} < b_{i} \leq \infty[/tex]
But if i look at the interval (0,1] in R, then its compliment is
[tex](-\infty,0] \cup (1,\infty)[/tex]
Which is a union of intervals of the real line, not a disjoint union. A disjoint union would seem to have sets of the form [tex](-\infty,0] \times {{0}}[/tex] which don't belong to the real line at all.
Where am I going wrong?
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