Past exam of the options I course of the University of Oxford / 2012 / M / M2 / a / Solution

A joint probability density function must be measurable, nonnegative almost everywhere, and normalized:It determines the joint law by $\mathrm{Pr}((X,Y)\in B)={\int }_{B}f(x,y)dxdy$ for measurable sets $B$. Values on sets of planar measure zero may be changed without changing that law.