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I would like to know the full connection to measure theory. It seems if I have a probabilistic measure space <math>(A,\mathcal{A},P_A)</math> and a measurable function <math>X\colon A\to \mathbb{R}</math> it might be the pushforward measure <math>X_* P_A= P_A \circ X^{-1}</math> which I guess would be a mapping <math> (\mathbb{R}, \mathcal{B}(\mathbb{R}), ||\cdot||_2)\mapsto \mathbb{R}</math> which for some reason is then restricted to half open sets <math>F_X \overset{?}{:=} (X_* P_A)\restriction [-\infty , r)\times \mathbb{R}</math>. But this is original research and feels a bit patchy (unclear to me how to generalize to random variables with values in <math>\mathbb{R}^n</math>) so please if someone who has connected the dots could add the connection. <!-- Template:Unsigned --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Rostspik|Rostspik]] ([[User talk:Rostspik#top|talk]] • [[Special:Contributions/Rostspik|contribs]]) 05:58, 27 April 2019 (UTC)</small> <!--Autosigned by SineBot-->
h <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2.183.8.104|2.183.8.104]] ([[User talk:2.183.8.104#top|talk]]) 05:39, 16 November 2019 (UTC)</small> <!--Autosigned by SineBot-->
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