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Math Help - need help on Borel algebra

  1. #1
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    need help on Borel algebra

    Hi,
    I've a quick question about Borel algebra. The Borel algebra on set X is defined as the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Let R be the set of reals and B(R) denote the Borel algebra on R. Then my question is: whether B(R)*B(R) equals B(R2)? How do you prove or disprove this?

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  2. #2
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    Re: need help on Borel algebra

    Could you remind what B(R)*B(R) is?
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  3. #3
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    Re: need help on Borel algebra

    By B(R)B(R), I mean the product of two Borel algebras.
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  4. #4
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    Re: need help on Borel algebra

    And could you remind what a product of two sigma-algebras is?
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  5. #5
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    Re: need help on Borel algebra

    Never mind. I found the answer here:
    Borel Sets on R^n - MathOverflow
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  6. #6
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    Re: need help on Borel algebra

    As the answer in link says, the set A=\{X_1\times X_2\mid X_1,X_2\in B(\mathbb{R})\} is a subset of B(\mathbb{R}\times\mathbb{R}). If B(\mathbb{R})\times B(\mathbb{R}) is defines as A, then B(\mathbb{R})\times B(\mathbb{R})\subsetneq B(\mathbb{R}\times\mathbb{R}), but if B(\mathbb{R})\times B(\mathbb{R}) is defined as the sigma-algebra generated by A, as the comments in the link suggest, then B(\mathbb{R})\times B(\mathbb{R})= B(\mathbb{R}\times\mathbb{R}).
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