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This appeared as a throwaway statement in a proof - that a strictly monotonic (increasing) transformation of a continuously distributed random variable (I am assuming that this means that the distribution function is continuous, not that the random variable is absolutely continuous) is also a continuously distributed random variable.

So the setup $X:\left(\Omega, \mathscr{F}, \mathbb{P}\right) \longmapsto \left(\mathbb{R}, \mathscr{B}(\mathbb{R}), \mathbb{P}_X\right)$ and $h: \mathbb{R} \longmapsto \mathbb{R}$ and $h(X) = Y$, where clearly $h$ needs to be measurable. So the claim is that if $h$ is monotonic, then $Y$ is a continuously distributed random variable.

A proof or a reference to a textbook would be appreciated.

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    What if $h$ is constant?2011-11-02
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    That is a good point; my fault, since I meant strict monotonicity.2011-11-02
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    Doesn't "$h$ is monotonic" imply $h$ is measurable (so that doesn't need to be a separate hypothesis)?2011-11-02
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    Yes, that was my (redundant) addition to the throwaway statement.2011-11-02

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