In probability textbooks, most theorems and statements start with something like "Let $(\Omega, \mathcal{A}, P)$ be a probability space". Now, I am wondering if there is some intuition from applications, e.g., in statistics, what distribution $P$ could be in many cases? So as an example lets say we have random variable $X$ that follows a Gaussian distribution $P_X$, which is the pushforward measure of underlying probability measure $P$. Can we now say anything about what distribution $P$ is?
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- 2$\begingroup$ You want to reconstruct the probability space $(\Omega,\mathcal{A},P)$ from the the push-forward $(\Omega',\mathcal{A}', P^X)$ where $X: \Omega \to \Omega'$ is a random variable? This is not possible. $\endgroup$psl2Z– psl2Z2025-10-21 10:06:15 +00:00Commented Oct 21 at 10:06
- $\begingroup$ For any random real variable, we can define certainly define a sample space with $\Omega'=\mathbb R.$ But it is far from the only sample space. If your space is an infinite sequence of die rolls, and $X$ is the sum of the second and third die, then the space with $\Omega'=\mathbb R$ is very tiny and simple compared to the "actual" space. $\endgroup$Thomas Andrews– Thomas Andrews2025-10-21 11:49:26 +00:00Commented Oct 21 at 11:49
- $\begingroup$ That‘s neither possible nor necessary: just use $\mathbb{R}$, Borel sets and the „pushforward measure“ as your probability space. $\endgroup$wasn't me– wasn't me2025-10-21 11:51:12 +00:00Commented Oct 21 at 11:51
- $\begingroup$ So its not that I want to reconstruct $P$. But my question is rather if we ever explicitely state how the underlying measure $P$ is definied? Or if there is a universally applicable choice or sth. Or one that covers most of the interesting cases? $\endgroup$guest1– guest12025-10-21 11:58:53 +00:00Commented Oct 21 at 11:58
- 1$\begingroup$ In a concrete scenario, an appropriate sample space Ω, σ-algebra A, and probability measure P would hopefully be suggested by the problem at hand. For example, if you model a coin toss, the obvious sample space to use is a two-element set Ω={H,T}. But you can choose another if you want. If you're working with a random variable with some given (pushforward) distribution already, the choice of underlying probability space is rarely important, and may typically be left unspecified. Or, as @wasn'tme suggests, just use $(\mathbb R,\mathscr{B}(\mathbb R),\mathrm{d} x)$ to start. $\endgroup$Rolf Alien– Rolf Alien2025-10-21 16:58:05 +00:00Commented Oct 21 at 16:58
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1 Answer
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3 Whenever you have a probability law $\mu:\mathscr{B}(S)\to[0,1]$ on a complete separable metric space $S$ equipped with its Borel sets $\mathscr{B}(S)$, a "natural" choice for the original probability space is arguably $(\Omega,\mathscr{F},P)=([0,1],\mathscr{B}[0,1],\lambda)$ where $\lambda$ is the Lebesgue measure. This is because of the representation theorem of Skorokhod: given such $\mu$, there exists a Borel measurable function $X:[0,1]\to S$ (a random variable) such that $\mu$ is the law of $X$. So whenever you work with a probability measure on a complete separable metric space, you can always and safely introduce a random variable $X$ on the aforementioned probability space which has such measure as its law; so for example saying "Let $X$ be a real valued random variable on $(\Omega,\mathscr{F},P)$ with distribution $P_X=$(...)" always makes sense, and formally the original space $(\Omega,\mathscr{F},P)$ can be identified with the unit interval.
- $\begingroup$ This seems far too abstract and remote from the level of the question. $\endgroup$Thomas Andrews– Thomas Andrews2025-10-21 15:41:54 +00:00Commented Oct 21 at 15:41
- $\begingroup$ @ThomasAndrews given OP's activity, I believe they are able to appreciate the insight $\endgroup$Snoop– Snoop2025-10-21 16:02:27 +00:00Commented Oct 21 at 16:02
- $\begingroup$ @Snoop Great answer thanks a lot! $\endgroup$guest1– guest12025-10-22 07:39:24 +00:00Commented Oct 22 at 7:39