If f is a continuous function, and ${X_n \to X} $ a.e then ${f(X_n) \to f(X)}$ a.e
probabilitymeasure-theory
asked 2012-10-17
user id:42826
45
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I think it would be helpful to include the space, sigma-algebra and measure under consideration. – 2012-10-17
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If $X_n(\omega)\to X(\omega)$ for $n\to\infty$ then by continuity $f(X_n(\omega))\to f(X(\omega))$ for $n\to\infty$. – 2012-10-17
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@Lord_Farin What for? – 2012-10-19
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Because it seems hard to believe that things can be said about arbitrary measure spaces built on arbitrary topologies. At least some assumptions need to be made. The sigma-algebra may have nothing to do with the topology. – 2012-10-19
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@Lord_Farin The sigma-algebra and the measure indeed have nothing to do with a topology. Random variables $X$ usually are measurable functions $X:\Omega\to\mathbb R^n$ where $\Omega$ is endowed with some sigma-algebra $\mathcal F$ and $\mathbb R^n$ is endowed with the Borel sigma-algebra $\mathcal B(\mathbb R^n)$. Hence at least the sigma-algebra $\mathcal F$ of the source-set $\Omega$ is irrelevant. Note also that the result holds irrespectively of the probability measure on $(\Omega,\mathcal F)$. (Unrelated: please use @.) – 2012-10-20
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@did: Confusion arose from me being more into abstract measure theory. The only reference to the fact that we are dealing with probability theory is that there is a tag "probability". Furthermore, your point is disproved by assuming the Borel $\sigma$-algebra, which arguably relies on the Euclidean topology on $\Bbb R$, in which (topological) sense the function is continuous. There's too many gaps of information left for the reader to decide - apparently you had less problems with guessing how to fill these gaps. Even that $X$ is a RV is not mentioned... Gaps gaps gaps. – 2012-10-20
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@Lord_Farin Agreed. (And I might re-use your *Gaps gaps gaps*, which I like very much. That is, if you don't mind.) – 2012-10-20