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Is it possible to define distribution function for a measure on a general measurable space, which generalizes the definition of distribution function for a probability measure on the extended real line?

If yes, how?

If no, how general/special can the measure and the measurable space be in order to define its distribution function?

Thanks!

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    Do you mean Radon-Nikodym?2010-11-17
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    I am not quite familiar with Radon-Nikodym, but would like to hear more.2010-11-17
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    Generally, the cdf on the line is a "substitute" for a corresponding Lebesgue-Stieltjes measure on the line. Elsewhere we use the measure only. Even on the line we can use the measure only, now that measure theory is well-known. Think of the cdf as a way to do things for people who don't know measure theory (either because they are non-mathematicians, beginning mathematicians, or mathematicians from many years ago).2011-07-12

2 Answers 2

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There is a well-known definition of cumulative distribution function $F$ for measures on finite dimensional Euclidean space ${\mathbb R}^d$. A function $F$ is the distribution function of a measure on ${\mathbb R}^d$ if and only if $F$ is right continuous and has non-negative increments, where these notions are defined appropriately for multidimensional space.

You can find details about this in Chapter 3 of Olav Kallenberg's Foundations of Modern Probability (2nd edition), for example.

I'm not aware of useful extensions of the idea of distribution functions to other partially ordered spaces, but they may exist.

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    Nice to know! Thanks!2010-11-17
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You can also define a measure via a "pullback construction". Let $(\Omega, {\mathcal S}, \mu)$ be a measure space. Suppose that $(Y, {\mathcal T})$ is a measurable space and $X: \Omega \rightarrow Y$ be a measurable function. You can define a measure $\mu_X$ by $$\mu_X(B) = \mu(X^{-1}(B)), \qquad B\in {\mathcal S}.$$ You can think of $\mu_X$ as being a "distribution measure. A change of variables formula is not hard to create.