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.
Let $B$ be the Borel measure on the real line, what is a good strategy to prove $B_{2}=B\times B$?
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real-analysis
1 Answers
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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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0I 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