General definition of characteristic function of a random variable and characteristic function of a probability measure

characteristic-functionsfourier transformprobability distributionsprobability theoryrandom variables

Let $(\mu_n)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$ and $(\hat{\mu}_n){_n\geq1}$ denote their Fourier transforms (aka characteristic functions).

Starting from the definition of characteristic function of a random variable $X$ (defined on a probability space $(\Omega\text{, }\mathcal{F}\text{, }\mu)$) as $$\varphi_X(u)=\hat{X}(u)=\mathbb{E}(e^{iuX})=\displaystyle\int e^{iux}d\mu(x)=\displaystyle\int e^{iux}\mu(dx)\tag{1}$$ one has that $$\hat{\mu}(u)=\displaystyle\int e^{iux}\mu(dx)\tag{2}$$

Two questions:

  1. As to the general definition of characteristic function of a random variable (see $(1)$), why does it hold that $$\displaystyle\int e^{iux}d\mu(x)=\displaystyle\int e^{iux}\mu(dx)$$?

    That is, how is it possible to show that $$d\mu(x)=\mu(dx)$$?

  2. As to the definition of the characteristic function of a probability measure (see $(2)$), as far as I understand I am integrating a function $f(x)$ with respect to probability measure $\mu$, with $x$ representing all the possible values of $\mu$. In other terms, I am integrating the function $e^{iux}$ with respect to the probability measure $\mu$ on the domain of the probability measure $\mu$ itself.

    Is that correct? Sorry if it could seem a bit messy.

Best Answer

$\int f(x)d\mu(x)$ and $\int f(x)\mu(dx)$ both denote the integral of function $f$ wrt probability measure $\mu$.

Using a more neutral notation that avoids the bound variable $x$ we could say that: $$\int f(x)d\mu(x)=\int fd\mu=\int f(x)\mu(dx)$$

For any fixed $u\in\mathbb R$ we have the (nice) function $f_u:\mathbb R\to\mathbb C$ that is prescribed by: $$x\mapsto e^{iux}$$Then:$$\hat{\mu}(u):=\int f_ud\mu$$is well defined for every probability measure $\mu$ on measurable space $(\mathbb R,\mathcal B)$.

If $X:\Omega\to\mathbb R$ is some random variable then it has a distribution $\mu$ on $(\mathbb R,\mathcal B)$ prescribed by $B\mapsto P(X\in B)$.

In that situation the function $\hat{\mu}(u)$ is labeled as "characteristic function of $X$".

This all remains valid if we work in $\mathbb R^d$ and with random vectors.

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