The importance of sigma-algebras

Main Question or Discussion Point

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 bc is also a member of A. Is this significant for some of the properties we want for a measure?

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jbunniii
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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.