Probability/Analysis Question

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

  • What does \sigma(X) mean here?

  • the σ-algebra generated by X

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

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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

  • thanks i'll check it out

The answer is accepted.