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So I am taking an intro to measure theoretic probability course and am a bit confused by how we defined sigma-algebras and measurable sets.

In particular if you have a sigma-algebra is that sufficient to guarantee that a corresponding probability measure is well defined on those sets? Or do we need more structure?

Moreover, in a sigma-algebra every element must have a well defined measure is that correct? I read here that the power set of the interval [0,1] is a sigma-algebra, but we know that the Vitali set is non-measurable, so I'm not sure how to reconcile that.

Even more confusing is the fact that if the power set of [0,1] is not a sigma-algebra then it is clearly not the case that you can define probabilities on any sigma-algebra, but it seems like in lectures we can.

So much confusion!

Any help would be appreciated.

Thanks,

Flint

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You are confusing two different structures here:

  1. A sigma-algebra is a set of sets. The axioms are something like "the union of countably many sets in the algebra must again be a set in the algebra" (and a few others). There are not numbers involved.
  2. A measure is a map from a sigma-algebra (the elements of which are then called "measurable sets") to the real (or sometimes complex) numbers. The axioms are something like "the measure of a disjoint union of countably many sets must be the sum of the individual measures" (and a few others).

So you do always need a sigma-algebra in order to define a measure, but the converse is not true. I.e. it is possible to have a sigma-algebra, where it is not possible to define a (non-trivial) measure on it. One example is the power set of $[0,1]$ which is a sigma-algebra, but it is not possible to have a measure on it. Also there can be more than one measure for the same sigma-algebra.

Edit: oh, just in case this wasn't clear: A probability-distribution is a special case of a measure.

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    $\begingroup$ A real measure is usually allowed to take the value $\infty$. For instance in the usual Borel $\sigma$-algebra over the reals (for the Lebesgue measure), we need to assign infinite measure to some sets. $\endgroup$ – egreg Feb 28 '17 at 16:40
  • $\begingroup$ Thanks for this! It's all clear now. $\endgroup$ – CAPM Mar 1 '17 at 0:32
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Simon's answer is very good. I just want to add that I very much doubt that the $\sigma$-algebra you're using in lectures is the whole of the power set of $[0,1]$. That would be useless in a probability course, since (as far as I'm aware) you can't define non-trivial measures on it.

Perhaps your lecturer is actually using the Lebesgue $\sigma$-algebra? This $\sigma$-algebra contains almost every set you'll ever encounter in daily life - including every possible countable union of points and/or intervals. The Lebesgue $\sigma$-algebra does however exclude some pathological sets - like the Vitali set. These pathological sets need to be excluded in order for it to be possible to define useful measures respecting the countable additivity property. And you don't really care about chucking these sets out, because, like I said, you hardly ever encounter them daily life!

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  • $\begingroup$ Yeah that is correct we are not using the power set of [0,1], but he just noted that it was an example. Thanks a lot for the clarification! $\endgroup$ – CAPM Mar 1 '17 at 0:30

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