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In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions."

  • The distribution function of a finite Borel measure $\mu$ on $\mathbb{R}$ is defined by $F(x) = \mu((-\infty, x])$.

  • The distribution function of a measurable function $f\colon X \to \mathbb{R}$ on a measure space $(X, M, \mu)$ is a function $\lambda_f\colon (0,\infty) \to [0,\infty]$ given by $\lambda_f(\alpha) = \mu(\{x\colon |f(x)| > \alpha\})$.

  • A distribution on an open set $U \subset \mathbb{R}^n$ is a continuous linear functional on $C^\infty_c(U)$.

Is there any sort of relationship between these concepts?

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2 Answers 2

up vote 4 down vote accepted

The first two uses came from probability theory, and are somewhat related as terminlogy. They are however, as far as I know, not related at all with the third concept.

In particular, given a measurable map $f:(X,M)\to(\mathbb{R},\mathcal{B})$, and a measure $\mu$ on $(X,M)$. $f$ (or I guess $|f|$ in your case) defines a push-forward measure $f_*\mu$ on $(\mathbb{R},\mathcal{B})$ by the definition that, for every $b\in\mathcal{B}$ the Borel sigma-algebra, $f_*\mu(b) = \mu(f^{-1}(b))$. Then the distribution function $\lambda_f$ is something like $1- F$ for the distribution function corresponding to the pushforward Borel measure $|f|_*\mu$. (The 1 should be replaced by the total mass of the measure $|f|_*\mu$ when it is not a probability measure.)

See the website Earlist Known Uses of some of the Words of Mathematics for some references for what I write below.

Now, the distribution in the sense of the continuous linear functional is introduced by Laurent Schwartz, in French. In French, however, the distribution function of your Borel measure (or of your measurable function) is called "la fonction de répartition", which strongly suggests that Schwartz's choice of terminology is completely independent of the probability/measure theory uses of the words.

In German, the language in which "distribution functions" were introduced, the probabilistic concept is Verteilungsfunktion, while the functional analytic concept is just taken straight from French/English as Distribution.

The above all strongly indicates that while senses 1 and 2 are related, they are disjoint for the 3rd use of the word distribution. In fact, English is one of the (perhaps few) unhappy languages in which they coincide.

(Just to confuse you further, there is also a use of the word distribution in differential geometry, which also means something completely different and disjoint from the three senses you listed above.)

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+1 for explanation of terminology in different languages –  mpiktas Feb 2 '11 at 13:23

Let's consider the relationship between the first two notions. It is convenient and natural to consider this in a probability setting, so that the total measure is $1$.

First notion. Suppose that $X$ is a random variable on a probability space $(\Omega,\mathcal{F},P)$. Define $\mu$ as follows. For any $B \in \mathcal{B}(\mathbb{R})$, $\mu(B)=P(\lbrace\omega : X(\omega) \in B\rbrace)$. The right-hand side is written in short as $P(X \in B)$. Then $\mu$ is a probability measure on $\mathcal{B}(\mathbb{R})$, which we call the distribution of $X$. Now, the function $F:\mathbb{R} \to [0,1]$ defined by $F(x) = \mu((-\infty,x])$ is called the distribution function of $X$. Note that $F(x) = P(X \in (-\infty,x]) = P(X \leq x)$, as one would expect. Of course, $F$ satisfies all the usual properties from probability theory.

Second notion. Let's first change the notation, in accordance with the previous notion, as follows. The distribution function of a random variable (that is, a measurable function) $X: \Omega \to \mathbb{R}$ on a probability space (that is, a measure space with total measure $1$) $(\Omega,\mathcal{F},P)$ is a function $\lambda_X:(0,\infty) \to [0,1]$ given by $\lambda _X (\alpha ) = P(\lbrace \omega :|X(\omega )| > \alpha \rbrace )$. As before, the right-hand side is written in short as $P(|X| > \alpha)$.

The relationship between the first two notions is thus established (in the setting of probability spaces). Specifically, $$ \lambda _X (\alpha ) = P(|X| > \alpha) = P(X > \alpha) + P(X < -\alpha) = [1 - F(\alpha)] + F(-\alpha^-), $$ where $F(-\alpha^-)=\lim _{s \uparrow -\alpha } F(s)$ (recall that $F$ is right-continuous with left limits).

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