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Here $B_{2}$ is the $\sigma$-algebra generated by open sets on the plane, $B$ is the $\sigma$-algebra generated by open sets on the real line. I need to prove the product measure coincides with the given measure.

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Open sets in the plane are generated by open balls (well, discs, but this arguments apply not only to $\mathbb R^2$ but to $\mathbb R^n$). The product topology is generated by open rectangles.

So what you want to prove is that every ball contains a rectangle, and that every rectangle contains a ball.

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    I found this construction is actually quite useful, because later I was asked to prove if two measure coincide on retangles, then it coincides on the whole $\sigma$-algebra.2012-12-03