# Inequality between product measure and its projection


Does this make sense?

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Just so you know, $X\times Y\setminus A\times B\neq (X\setminus A)\times (Y\setminus B)$ in general. –  Clayton Mar 19 '13 at 0:55

Well, it make sense and actually it's even simpler: note that $$X\times Y\setminus A\times B \subseteq [(X\setminus A)\times Y]\cup[X\times (Y\setminus B)]$$ which if not obvious, may follow with the help of these pictures: $$X\times Y\setminus A\times B$$ $$(X\setminus A)\times Y$$ $$X\times (Y\setminus B)$$
As a result you get $$\gamma(X\times Y\setminus A\times B)\leq \mu(X\setminus A) + \nu(Y\setminus B)$$