In probability theory, how may we define a law (without need of defining a random variable first)?

Question

Let $(\Omega, \mathcal{F}, P)$ be a probability triple. The following definitions are from the third chapter of William's Probability with Martingales.

(Definition of a distribution function and law given a random variable)
Definition: given a random variable $X$, we define its distribution $F_X:\mathbb{R}\to[0,1]$ and law $\mathcal{L}_X:\mathcal{B}\to[0,1]$ (which is a probability measure) by \begin{equation} \label{rel rv d l} F_X(c) := \mathcal{L}_X(-\infty,c] := P(X\le c) = P\{\omega :X(\omega)\le c\}. \end{equation}

(Definition of a distribution function in general)
Definition: a distribution function $F$ is a function $\mathbb{R}\to[0,1]$ such that
(a) $F$ is monotonically increasing.
(b) $\lim_{x\to -\infty} F(x) = 0$ and $\lim_{x\to \infty} F(x) = 1$.
(c) $F$ is right-continuous.

The two definitions above gives us the following: a function $F$ is a distribution function if and only if there is some random variable $X$ so that $F=F_X:=c\mapsto P(X\le c)$.

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With all this in mind, I wonder how may we define a Law $\mathcal{L}$ in general so as to make the following statement true:

A function $\mathcal{L}$ is a law if and only if there is some random variable $X$ so that $\mathcal{L}=\mathcal{L}_X:=(-\infty,c]\mapsto P(X\le c)$.

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Perhaps I should mention that there is a 'trivial' way of answering the question: a law $\mathcal{L}$ is defined as a function $\mathcal{B}\to[0,1]$ such that there exists a distribution $F$ with $F(c) = \mathcal{L}(-\infty,c]$ for any $c\in\mathbb{R}$. However, I'm looking for a definition of a law that does not mention a distribution either.

Answer 1

Edit: I'm actually not sure about my answer. Honestly, I can't believe I got 3 upvotes without comments that say I'm wrong or something. Guess that indeed means I'm right about at least 1 of the 2 things in Part 4. If you indeed think I'm correct, then please explain why you think so.

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Note that it's been awhile since I've this done this, so I may be rusty.

Part 1. A distribution function is defined as...

a right-continuous non-decreasing function $F: \mathbb R \to [0,1]$ satisfying $\lim_{x \to - \infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$.

Part2. Indeed every 'distribution function'...

(any $F$ that satisfies the above) is actually 'the distribution function' of some random variable $X$.

I believe an example of such $X$ is $X: (0,1) \to \mathbb{R}$ by $X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\}$ on the probability space $((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ is Lebesgue measure from here. Also see there. (I think this is to do with inverse distribution or quantiles.)

• Note: I don't think this is the only possible $X$. I mean yeah in general different random variables can have the same distribution like indicator random variables $1_A$ and $1_{A^C}$ in the same probability space $(\Omega, \mathcal F, \mathbb P)$ for any event $A$ of probability between $0$ and $1$. (Maybe even if 0 and 1, but it's a headache to think about this.) Hmmm...wait not sure...I think it's supposed to be $1_A$ and $1_B$ but $A$ and $B$ have the same probability. Ah whatever you do that $P(1_{\{\cdot\}} \le x)$ yourself.

Part 3. Law and distribution function

Distribution function of random variable $Y$ is $F_Y(y):=P(Y \le y)$ aaand law of $Y$ is $\mathscr L(Y \in B)$...related in that $\{Y \le y\} = \{Y \in (-\infty,y]\}$ and that $B \in \mathscr B := \sigma(\mathbb R) = \sigma(\pi(\mathbb R))$, where $\pi(\mathbb R) := \{(-\infty, y]\}_{y \in \mathbb R}$

Part 4. Law...

Oh I see now. You want to define law without random variable the same way we do distribution function? Well... Maybe

1. Any $\mathscr L: \mathscr B \to [0,1]$ s.t. $\mathscr L$ restricted to $\pi(\mathbb R)$ is a distribution function?

2. Any probability measure on $(\mathbb R, \mathscr B)$?

See this exercise in Ch3.9

My guess is that just as every law of a random variable is probability measure on $(\mathbb R, \mathscr B)$, we have that every probability measure on $(\mathbb R, \mathscr B)$ is also the law of some random variable. Maybe something that looks like $X(\omega) = \sup\{(?) B \in \mathscr B: \mathscr L(B) < \omega\}$ in $((0,1), \mathfrak{B}(0,1),\lambda)$...or just make the law into a distribution and make the random variable from that distribution.

Is there a way to go straight from law to random variable without distribution? Hmmm...Can't think of something, but I have a feeling there should be! If anyone knows, then please share!

Re your

'I'm looking for a definition of a law that does not mention a distribution either.'

• Right so refer to (2) in part 4. I think your

'such that there exists a distribution'

is my (1) in part 4, and I'm actually not so sure about that.

Anyhoo, I do look forward to learning from the corrections in the comments!

Answer 2

A "law" (in this context) is just a Borel probability measure on $\mathbb{R}$.

In real analysis, if $F:\mathbb{R}\to\mathbb{R}$ is nondecreasing and right-continuous, then there is a unique Borel measure on $\mathbb{R}$ such that $\mu((a,b])=F(b)-F(a)$. This measure is called the Lebesgue-Stieltjes measure associated to $F$. Two such functions produce the same measure if and only if they differ by a constant.

Conversely, if $\mu$ is a Borel measure on $\mathbb{R}$ that is finite on bounded sets, then it is a Lebesgue-Stieltjes measure of some nondecreasing and right-continuous function. One such function is $$ F(x) = \begin{cases} \mu((0, x]) &\text{if $x>0$},\\ 0 &\text{if $x=0$},\\ -\mu((x,0]) &\text{if $x<0$}. \end{cases} $$ For more details about the above correspondence, and about Lebesgue-Stieltjes measures, you can see Section 1.5 in Real Analysis: Modern Techniques and Their Applications by Folland.

From here, it is very easy to see that there is a one-to-one correspondence between distribution functions and Borel probability measures on $\mathbb{R}$. In other words, a function $\mathcal{L}$ that maps Borel subsets of $\mathbb{R}$ to $[0,1]$ is the law of a random variable if and only if $\mathcal{L}$ is a probability measure.