Let $X,Y$ be some nice measureable spaces (i'm interested in $[0,1]$ so we can assume compact, etc.). let $\mu$ be a measure on $X\times Y$.(again, assume it's nice, i.e. probability measure. anything else needed?
Are there necessarily measures $\mu_1,\mu_2$ on $X,Y$ resp. such that $\mu=\mu_1\times\mu_2$?
I thought it is a reasonable conjecture, but i couldn't prove it. My intuition is the lebesgue [/haar] measure, for which the answer is obviously true. I thought one could define $\mu_1(A)=\mu(A\times Y)$ and reps. $\mu_2$, but I couldn't show it would be a good construction (it is good for lebesgue, though).
EDIT - only this is now relevant:
thanks guys. But now i'm slightly confused, as I saw in a paper the terminology "projection of a measure", which is supposed to be some measure on XX derived from the measure on $X\times Y$. this must be something like the pushforward measure by projection, which is μ1μ1 that I defined here if i'm not wrong. I think that the paper does not really require the measure to be the product of those (which I now understand is not true), but perhaps satisfy some kind of a fubini identity $\mu(A)=\int\mu_1(A^y)d\mu_2(y)$? is that necessarily true?, (under nice enough conditions)
The goal is to deduce properties of μ from the projections, so what could be true?