Can Distributions of Combined Sample Spaces Be Derived from Individual Ones?

[mmzaj]

Dear all

I've just begun studying measure theory , and i can't help it but to think of it in terms of probability theory , i don't know if that is right or wring . any way , i have this naive question :

consider the following : we have n sample spaces $$\Omega_{}i$$, each with a distribution P$$_{}i$$ ( i=1,...n) , if we combine (union) the sample spaces to form a new sample space whose distribution is unknown , is there a way to extract the distribution of the new sample space from the previously know distributions ??

 

[EnumaElish]

The simplest case is to assume disjoint sample spaces. In that case, the probability of obtaining element x from the k'th space, x(k), will be P{x(k)} = P{x|k}P{k} = P_k{x}P{k}, where P{k} is the probability of obtaining the k'th space within the set of all spaces, or the measure of the k'th space in the union.

In general:

$$P\{x\} = \sum_{k=1}^N P\{x|k\}P\{k\}$$

where N is the number of spaces.

This is related to the axiom of independence [from] irrelevant alternatives in decision theory.

 

[mmzaj]

what about the general case ?

 

[EnumaElish]

See the portion of my post that begins with "In general."