How to understand joint distribution of a random vector?

1. Given a random vector, what are the domain, range and sigma algebras on them for each of its components to be a random variable i.e. measurable mapping? Specifically:
• is the domain of each component random variable same as the domain of the random vector, and are the sigma algebras on their domains also the same?
• Is the range of each component random variable the component space in the cartesian product for the range of the random vector? What is the sigma algebra on the range of each component random variable and how is it induced from the sigma algebra on
the range of the random vector?
2. Please correct me if I am wrong. If I understand correctly, given a random vector, the probability measure induced on the range (which is a Cartesian product space) by the random vector, is called the joint probability measure of the random vector. The probability measure induced on the component space of the range by each component of the random vector, is called the marginal probability measure of the component random variable of the random vector.
3. Consider the concept of the component random variables of a random vector being independent. I read from a webpage that it is said so when the joint probability measure is the product measure of the individual marginal probability measures. I was wondering if the sigma algebra for the joint probability must be the same as the product sigma algebra for the individual probability measures, or the former can just contain the latter?

Thanks and regards!

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I'll take a stab at answering these:

1. (Part a) Yes, the domain probability space of a random vector is the same as the domain probability space of its components. Think of a random vector as a vector-valued random variable.

I'm having trouble parsing these questions, because I can't tell whether you're using the word "range" to mean "image" or "codomain". I'll assume you mean "codomain".

1. (Part b) Given a probability space $\Omega$ and measurable spaces $E_1,\ldots,E_n$, a random vector is a random variable $\Omega \to E_1\times\cdots\times E_n$. Typically, each $E_i$ starts with a $\sigma$-algebra on it, and then the $\sigma$-algebra on $E = E_1\times\cdots\times E_n$ is defined as the product algebra. (That is, the $\sigma$-algebra on $E$ is generated by all products of the form $A_1\times\cdots\times A_n$, where $A_i$ is measurable in $E_i$ for each $i$.) Sometimes it is helpful to expand the $\sigma$-algebra on the product slightly, e.g. if you want some probability measure on the product to be complete.

2. That seems right. "Marginal probability measure" would also refer to the measure obtained on a product like $\prod_{i\in S} E_i$, where $S$ is some subset of $\{1,\ldots,n\}$.

3. I suppose it's fine for the $\sigma$-algebra on the product to be slightly larger than the product $\sigma$-algebra, e.g. if we want the measure on the product to be complete. However, the measure on the product should have the property that it is the unique extension of the product measure to the $\sigma$-algebra on the product. That is, the measure on the product should either be the product measure, or the completion of the product measure, or something in between.

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Thanks! You are right and by "range" I mean codomain. For 1. part b, I would like to know how to get the sigma algebra on each $E_i$ from a given sigma algebra on $E$, when getting each component random variable from the random vector? Thanks! – Tim Feb 14 '11 at 22:34