Minimum conditions for defining joint PDF

[winterfors]

Suppose one knows a probability density p(x) over a space X (where x\in X) and a conditional probability density p(y|x) over a space Y (where y\in Y).

This implies the integral \int{p(x)dx} is well defined as well as \int{p(y|x)dy}.

Defining a joint probability density

p(x,y)\ =\ p(y|x)p(x) ,​

p(x) will clearly be its marginal density over X, and the double integral

P(C)\ = \ \iint\limits_{(x,y)\in C}{p(x,y)dydx}​

is well defined for all measurable subsets C\ \subseteq \ X\times Y.One commonly assumes that integrals with reversed order of integration are equivalent

P(C)\ = \ \iint\limits_{(x,y)\in C}{p(x,y)dxdy}\qquad ,​

which also implies that the marginal probability density over Y exists and is uniquely defined p(y)\ = \ \int{p(x,y)dx}.This is not necessarily the case, since the change in order of integration poses restrictions on the integrand p(x,y). One sufficient condition, I believe, is that p(x,y) is continuous, but that is clearly not the minimal condition required. For instance, change of order of integration can be done for a p(x,y) that involves step functions, which are obviously not continuous.

Does anyone know what the minimal conditions are for changing the order of integration, in this case?

 

[gel]

If p>=0 then you just need to know that p is measurable. More generally, if p is measurable then
<br /> \int\int |p(x,y)|dxdy=\int\int |p(x,y)| dydx<br />
and, as long as this is finite then you can commute the integrals in \int\int p(x,y)dxdy. See http://en.wikipedia.org/wiki/Fubini%27s_theorem" .