1

This is a slightly soft question, but I am wondering if it is sensible to talk about the moments of a given measure $\mu$. In particular, suppose that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Is there a way to define $\mathbb{E}[\mu]$ for measures $\mu$ on $\mathbb{R}$ such that $\mathbb{E}[\mu_X] = \mathbb{E}_\mathbb{P}[X]$? Seeing as the expectation of a random variable is only dependent on its distribution, I am thinking that such a definition should be possible without reference to the underlying random variable.

If this definition is possible, can we then extend it to $n$th moments of a given measure? Could we also extend it (using the Bochner Integral) to arbitrary measures $\mu: B \to [0,\infty]$ where $B$ is a Banach space?

Harry Partridge
  • 697
  • $\int xd\mu$ does not make sense. Also, $\mu_X$ is a measure on $\mathbb R$, not on $\Omega$. – Kavi Rama Murthy Apr 17 '24 at 08:04
  • 1
    Most of the notations do not make any sense. To say $\mu << P$ you have to have $\mu$ defined on $\Omega$, so $\int xd\mu$ has no meaning. – Kavi Rama Murthy Apr 17 '24 at 08:18
  • 1
    Then you can only integrate functions from $\Omega$ to $\mathbb R$ w.r.t. $\mu$. How can you integrate the identity function on $\mathbb R$ w.r.t. $\mu$? – Kavi Rama Murthy Apr 17 '24 at 08:22
  • Ok I see that my formulation might not be quite right. My question is if such a definition is possible - I made an attempt at such a definition and it wasn't correct but I would appreciate it if you could help me to make a correct definition or show why such a definition is not possible. – Harry Partridge Apr 17 '24 at 08:27
  • 1
    In this form I think the answer is yes (somewhat trivially) - define $\Bbb E[\mu]$ to be the $\mu$-integral of the identity function on $\Bbb R$. This is a special case of this fact. For the same reason, you can define the $n$-th moment to be the $\mu$-integral of the function $x^n$. – Izaak van Dongen Apr 17 '24 at 10:01
  • 1
    In other words: Apart from the notation $\mathbb E[\mu]$ that is weird @IzaakvanDongen 's suggestion means nothing else that when $\mu$ is a measure on $\Omega=\mathbb R$ or $\Omega=\mathbb R^n$ it is a probability distribution of some $\mathbb R$-valued or $\mathbb R^n$-valued r.v. $X$ (which we can forget about). The expectation is then $\int x,d\mu$ and variance or moments are defined analogously. – Kurt G. Apr 17 '24 at 10:25

0 Answers0