# Definition of a semialgebra

1. Sep 3, 2009

### LeonhardEuler

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" [Broken]. 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
$$(-\infty,0] \cup (1,\infty)$$
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 $$(-\infty,0] \times {{0}}$$ which don't belong to the real line at all.

Where am I going wrong?

Last edited by a moderator: May 4, 2017
2. Sep 4, 2009

### MathematicalPhysicist

A disjoint union is a union of disjoint sets, so what you wrote is indeed a disjoint union of two intervals.

3. Sep 4, 2009

### LeonhardEuler

What about what it says in those two links? I have not seen that definition of a disjoint union.

4. Sep 4, 2009

### g_edgar

Ignore that Wikipedia page. It is irrelevant for this problem. In this problem, a "disjoint union" of sets is a union of sets where the sets are disjoint.

5. Sep 4, 2009

### LeonhardEuler

Well that makes a lot more sense then. The last part that confuses me is that I can only write the compliment of that set with an interval that extends to +infinity which is right open, not right closed. How is this problem resolved.