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Let $\mu$ be a positive finite measure on a $sigma$-algebra in $X$. Let's consider the product measure $\nu:=\mu \times \mu$ in $X \times X$. Is it true that for every measurable subset $E\subset X\times X$ we have $\nu (E)=inf [ \mu(A)\times \mu(B): E \subset A\times B, \ A, B \ measurable ]$ and $\nu (E)=sup [ \mu(A)\times \mu(B): A\times B \subset E, \ A, B \ measurable ]$.

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I don't think this question is research-level, but here we go.

The answer is no. For example, let $\mu$ be Lebesgue measure on the interval $[0,1]$ (with the Borel $\sigma$-algebra.) Let $D=\{(x,x):x\in [0,1]\}$ be the diagonal in the square. If $D\subset A\times B$, then $A$ and $B$ must both contain $[0,1]$, so $\nu(D)=0$ but the infimum in your right-hand side is $1$.

To show the other equality is also false you can take the same measure and $E$ to be a small neighborhood of the diagonal.

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