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

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SUMMARY

The discussion centers on the transformation of random variables within the context of probability spaces, specifically focusing on the measurable map from a random variable X to another space T. The necessity of restricting to σ-finite measures on the measurable spaces S and T is highlighted, as it relates to the existence of density functions. The participants agree that understanding these measures is crucial for accurately determining the distribution of the transformed random variable r(X) based on the distribution of X.

PREREQUISITES
  • Understanding of probability spaces, specifically the components (Ω, F, P).
  • Familiarity with measurable spaces and measurable functions.
  • Knowledge of random variables and their transformations.
  • Concept of σ-finite measures and their significance in probability theory.
NEXT STEPS
  • Study the properties and applications of σ-finite measures in probability theory.
  • Explore the concept of density functions and their role in random variable transformations.
  • Learn about the implications of measurable maps in the context of probability distributions.
  • Investigate the relationship between random variables and their distributions in various measurable spaces.
USEFUL FOR

Mathematicians, statisticians, and students studying probability theory, particularly those interested in the transformation of random variables and the implications of σ-finite measures.

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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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