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I'm having trouble understanding how the definition of a measurable function relates to the notion of random element and random variable. The definition I have at hand states the following:

Let $(\Omega, \mathcal{F})$ and $(\Lambda, \mathcal{G})$ be measurable spaces and $f: \Omega \rightarrow \Lambda$. Then $f$ is a measurable function if and only if $f^{-1}(\mathcal{G}) \subset \mathcal{F}$.

I thought I understood this definition as I read it. Translated to less rigorous language, it states that, if there is a set of events $w \in \Omega$ such that $f(w)= \mathcal{G}$ and $w \in \mathcal{F}$, then $f$ is a measurable function. Even more plainly, if the whole $\sigma$-field of $\Lambda$ is a subset of $\mathcal{F}$, then $f$ is a measurable function if it maps that subset to that $\sigma$-field.

However, right after the definition it is stated that, in probability theory, $f$ is called a random element and denoted $X, Y, Z...$, and that if $X$ is measurable from $(\Omega, \mathcal{F})$ to $(\mathcal{R}, \mathcal{B})$, then it is called a random variable. And here is when I feel lost.

How is the fact that $X$ is measurable from $(\Omega, \mathcal{F})$ to $(\mathcal{R}, \mathcal{B})$ related to what we understand, in probability and statistics, as a random variable? Why is the function that relates some subset of $\mathcal{F}$ to the $\sigma$-field of $\Lambda$ a 'random element'?

Note. I know similar questions have been answered here and here. However, the answers did not make emphasis on the notion of randomness, which is the one I am confused, so they did not help me.

lafinur
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2 Answers2

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You write:

I thought I understood this definition as I read it. Translated to less rigorous language, it states that, if there is a set of events $w \in \Omega$ such that $f(w)= \mathcal{G}$ and $w \in \mathcal{F}$, then $f$ is a measurable function. Even more plainly, if the whole $\sigma$-field of $\Lambda$ is a subset of $\mathcal{F}$, then $f$ is a measurable function if it maps that subset to that $\sigma$-field.

No. That is not at all a correct translation. First, the notation is wrong. If $\omega\in\Omega$ then $f(\omega)\in \Lambda$. Note that $\mathcal{G}$ is a collection of subsets of $\Lambda$ (i.e a set of sets). The definition of measurability is what it is: preimages of every set in $\mathcal{G}$ is in $\mathcal{F}$. More explicitly, the condition is that

For every set $G\in \mathcal{G}$, we require that the set $\text{preimage}_f(G)\equiv f^{-1}(G):= \{\omega\in \Omega\,:\, f(\omega)\in G\}$ belongs to $\mathcal{F}$. i.e $f^{-1}(G)\in \mathcal{F}$.

Setting that aside, I think you're unnecessarily getting confused by probabilists terminology. We use "variable" when the target space is $\Bbb{R}$. When the target space is just some arbitrary set $\Lambda$, we just say "a random element", or more properly "a random variable in $\Lambda$". Or the most unambiguously, just a function from $\Omega$ to $\Lambda$ (though probabilists don't like to use the function terminology).

So, random variable, random element, random vector, random graph, random matrix are all just certain functions from one set into another set, satisfying the condition of measurability (with respect to appropriate $\sigma$-algebras). The word following random is just to be more specific about what elements of the target space of the function are. See this answer for a more detailed explanation.

So, to summarize

  • "random" is referring to the fact that you're dealing with functions $f:\Omega\to \Lambda$, and not just a single element $\lambda_0\in \Lambda$.
  • The word following "random" is meant to make things more specific.

Finally, for technical reasons we add in the extra condition that the function we deal with satisfy the condition known as "measurability".

peek-a-boo
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You are right that there is no randomness in the definition. The randomness comes when you have a probability measure $P$ on your measurable space $(\Omega, \mathcal{F})$. The thing of most interest is the distribution of random variables. The distribution of a random variable $X$ is the measure $P_X$ on $B(\mathbb{R})$ defined by $P_X(A) = P(X \in A) = P(X^{-1}(A))$. So in statistics, when you say "X is normally distributed", what you mean is that $P_X$ is the normal distribution on $\mathbb{R}$. Of course, you don't have to limit yourself to $\mathbb{R}$-valued random variables. The same things work when $X$ is $\mathbb{R}^n$ valued. In this case, $P_X$ is a measure on $B(\mathbb{R}^n)$.

Mason
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