Naked before God: HELP ME SKLANSKY!

[jason1990]

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If we can assume that X and Y are never equal

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Is it sufficient to assume that the probability that x=y is zero?

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I believe it is necessary and sufficient to have the CDF of X be continuous, which happens if and only if f(y):=P(X=y)=0 for all y. Do a little logical song and dance to show that f(y)=0 for all y if and only if f(Y)=0 almost surely. Then note that since f(Y) is nonnegative, f(Y)=0 almost surely if and only if E[f(Y)]=0. Finally, observe that

P(X=Y) = E[P(X=Y|Y)] = E[f(Y)].

So the CDF of X is continuous if and only if P(X=Y)=0.