Math Is Fun Forum

That was a dead-on analysis of my confused mind!  I want you to run through my thoughts to see if I'm getting it alright.  Here we go... (I've used the definition of measure from Wikipedia)

A measure μ is a function that is defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied:

1) The empty set has measure zero
2) Countable additivity or σ-additivity: if E1, E2, E3, ... is a countable sequence of pairwise disjoint sets in Σ, the measure of the union of all the Ei is equal to the sum of the measures of each Ei.

Why is the measure defined on a σ-algebra, and not on any arbitrary set?

The properties  of a σ-algebra facilitate the fulfillment of the conditions placed on any measure:

1) A σ-algebra guarantees the existence of the empty set and hence the measure of the empty set can be defined.  It depends on the application to assign a value of zero to the measure of the empty set for condition (1) to be satisfied.

2) The property of a σ-algebra being closed under countable unions guarantees that the sum of the measures of a countable number of disjoint sets in Σ can be represented as the measure of another set (which will be the result of the countable unions) in Σ.  Whether the condition (2) is satisfied or not solely depends on the way the measure is defined for the various (elementary) sets in Σ.

Here are some lingering doubts:

1) Is there any more relationship between σ-algebra and measure that is basic, and cannot be derived?

2) Suppose I look at a probability problem and construct a sample space Ω and assign probabilities to all the elementary events in Ω, satisfying the properties of μ(Ø) = 0 and sum of all elementary events being equal to 1, i.e. μ(Ω) = 1.  Does it mean that I am safe now and wouldn't have to worry about whether an event (a subset of Ω) is measurable or not?  How about the case of an infinite sequence of events?  Are there any concerns lurking out there?

Thanks Ricky!