The importance of sigma-algebras

[aaaa202]

Why is it so important that a measure is defined on a sigma-algebra? Which properties of the sigma-algebra are crucial for the properties of a measure? Since a measure is by axiom defined on countable unions of sets, it makes sense that a measure should be defined on a family of sets which preserves this property. But for a sigma-algebra A we also have as axiom that if b is a member of A then b^c is also a member of A. Is this significant for some of the properties we want for a measure?

 

[jbunniii]

Here are a few reasons we want the complement of a measurable set to be measurable:

1) If I divide a measurable object into two pieces, it would not be very convenient if one piece was measurable and the other was not.

2) If there is a topology on the space, we generally want both the open and the closed sets to be measurable, and these are complements of each other.

3) For a probability measure, we often need to use the relation $P(X) + P(X^c) = 1$.

4) For measurable functions, it's useful to be able to measure the sets $\{x : f(x) \leq c\}$, $\{x : f(x) > c\}$, $\{x : f(x) < c\}$, and $\{x : f(x) \geq c\}$. The first two are complements of each other, and the second two are complements of each other.