Probability/Analysis Question

Suppose $X : R \rightarrow R$ is a Borel random variable, and suppose that the image of X is in Q. Show that $\sigma(X)$ is not the whole Borel $\sigma$-algebra.

Probability Real Analysis
L Ellis L Ellis
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  • Erdos Erdos
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    What does \sigma(X) mean here?

  • L Ellis L Ellis
    0

    the σ-algebra generated by X

  • Erdos Erdos
    0

    σ(X) is the smallest sigma algebra F such that X is a measurable function? I see.

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Erdos Erdos
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  • Erdos Erdos
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    For Doob's Theorem, See page 37 of the following textbook: https://math.aalto.fi/~kkytola/files_KK/ProbaTh2019/ProbaTh-2019.pdf

  • L Ellis L Ellis
    0

    thanks i'll check it out

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