I On transformation of r.v.s. and sigma-finite measures

AI Thread Summary
The discussion focuses on the transformation of random variables and the necessity of using σ-finite measures in the context of probability spaces and measurable functions. It highlights the goal of finding the distribution of a transformed random variable, r(X), based on the distribution of the original variable, X. The mention of σ-finite measures suggests a requirement for certain mathematical properties, likely related to the existence of density functions. The need for σ-finite measures is questioned, indicating a desire for clarity on its implications in the transformation process. Understanding this concept is crucial for accurately analyzing the distribution of transformed random variables.
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I'm reading an article on transformation of random variables. In the article they restrict to ##\sigma##-finite measures, but I don't understand why.
I'm reading this article on transformation of random variables, i.e. functions of random variables. We have a probability space ##(\Omega, \mathcal F, P)## and measurable spaces ##(S, \mathcal S)## and ##(T, \mathcal T)##. We have a r.v. ##X:\Omega\to S## and a measurable map ##r:S\to T##. Then we want to find the distribution of ##r(X)## given that of ##X##. Pretty soon into the article, after the first proposition, under the very first diagram, they say that we should then consider ##\sigma##-finite measures on ##S## and ##T##. I don't understand why we need to restrict to ##\sigma##-finite measures. What necessitates this?

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Ok, I guess you can ignore the question. I believe it is because of the existence of density functions, if I'm not mistaken.
 
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