Distribution and Probability Distribution

Question

I'm studying on the book of Kolmogorov and Fomin: "Elements of the Theory of Functions and Functional Analysis". I'm into the measure theory and I finished the Theorem of Radon-Nikodim. Now finally I understood why $$P(A)=\int_{A} f(x) dx$$ where, if I'm not wrong,$dx$ should be $d\mu$, or the Lebesgue-measure. Now we call it probability distribution. At page 204 of the book it says that a REGULAR distribution can be written in this form: $$T(\varphi)=\int_{-\infty}^{\infty} f(x)\varphi(x) dx$$ I don't understand how I can write the probability distribution in this form, did I miss something? Or probability distribution are not "distribution" as I think.

Answer 1

There's two different "distributions" in your question:

1. Probability distribution $P$ on $\mathbb{R}$. This is a measure on $\mathbb{R}$. If $P$ is continuous with respect to the Lebesgue measure $d \mu$, then we can write it as $$\mu(A) = \int_A f(x)d\mu (x)$$
2. Distributions as in "generalized functions". These are continuous linear functionals on some function space, i.e. maps $T : \mathcal{D} \to \mathbb{R}$. Usually $\mathcal{D}$ is the set of $C^\infty$ functions with compact support. If $f$ is a locally integrable function then $$T(\phi) = \int_{-\infty}^\infty f(x) \phi(x) dx$$ is an example of such a distribution. The distributions that can be written in this way are called regular distributions. Another example of a distribution is the dirac delta.

If you have a probability distribution $P$, you can get a linear functional on $C^\infty_0$ by integrating w.r.t. the probability distribution. If $P$ is continous with respect to lebesgue measure, then the linear functional you get will be a regular distribution, by Radon Nikodym.