Simplifications assuming function is a probability distribution

Question

Is it possible to define a generic probability distribution function (pdf) so Mathematica can use its properties to simplify expressions?

For example:

Simplify[Integrate[f[x],{x,-Infinity,Infinity}]]

How to tell to Mathematica that f[x] is a PDF so it can simplify (in this case) to 1?

This is easy for a PDF from a given distribution, but my question is about the case of a generic PDF f[x]

Answer 1

ProbabilityDistribution does precisely that.

https://reference.wolfram.com/language/ref/ProbabilityDistribution.html

Answer 2

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

Use TagSet or TagSetDelayed to associate the integral to the symbol f, e.g.,

f /: Integrate[f[x_], {x_, -Infinity, Infinity}] = 1;

Then

Integrate[f[y], {y, -Infinity, Infinity}]

(* 1 *)

EDIT:

Mathematica doesn't combine integrals unless explicitly told to do so. Define a rule

intRule = Integrate[expr_, {t_, a_, b_}] +
    Integrate[expr_, {t_, b_, c_}] :>
   Integrate[expr, {t, a, c}];

Applying the rule,

Integrate[f[y], {y, -Infinity, 10}] + 
  Integrate[f[y], {y, 10, Infinity}] /. intRule

(* 1 *)