U**R
发帖数: 5 | 1 First of all, happy new year to everyone!
I'm stumbled by a question that, I believe, ought to be easy...
Given two measure spaces (X, Bx, p) and (Y, By, q), each equipped with a
sigma-finite measure, p on the sigma-field Bx over X and q on the sigma-
field By over Y, respectively, we know that there exists a product measure
space (X*Y, Bx@By, p@q), where "*" denotes Cartesian product and "@" denotes
tensor product. Suppose, furthermore, that there are two other sigma-finite
measures, P on Bx ov | s******h
发帖数: 539 | 2 (1) => (2), I'll give you the hint:
By uniqueness of measures, we only have to consider sets from
\{A\timesB: A\in B(X), B\in B(Y)\}, then use Tonelli's theorem, we can show
that p\times q << P\times Q, with the corresponding R-N derivative
dp/dP(x)*dq/dQ(y) with respect to the product measure [P\times Q](x, y).
Here is an intuitive example:
If you have two independent continuous random variables X, Y such that X~N(0
,1), Y~Cauchy(0,1). Then the joint probability measure of X and Y is
dominated |
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