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.
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.
See the attached file.
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