In our reading we first defined the characteristical function of a probability mesaure as follows:
Let $\mu$ be a probability measure on $(\mathbb{R},\mathcal{B})$. The Fourier transform $\varphi_{\mu}\colon\mathbb{R}\to\mathbb{C}$, $$ \varphi_{\mu}(t):=\int e^{itx}\, d\mu(x) $$ is called the characteristical function of $\mu$.
Then we defined what the characteristical function of a random variable is:
Let $X$ be a random variable with $\mathbb{P}(\left\{|X|<\infty\right\})=1$. Then $\varphi_X:=\varphi_{\mathbb{P}_X}$ is called the characteristical function of $X$. It is $$ \varphi_X(t)=\int e^{itx}\, d\mathbb{P}_X=\mathbb{E}(e^{itX}). $$
(With $\mathbb{P}_X$ we always meant the distribution of $X$, i.e. $\mathbb{P}\circ X^{-1}$.)
My questions are:
(1) Why is $\mathbb{P}(\left\{|X|<\infty\right\})=1$ given as condition resp. why is it necessary?
(2) Why is $\int e^{itx}\, d\mathbb{P}_X=\mathbb{E}(e^{itX})$?
Edit
Concerning (2) $$ \int e^{itx}\, d\mathbb{P}_X=\int e^{itx}\circ X\, d\mathbb{P}=\int h_t\circ X\, d\mathbb{P} $$ with $h_t\colon\mathbb{R}\to\mathbb{C}, x\mapsto e^{itx}$. It is $$ h_t\circ X\colon\Omega\to\mathbb{C}, \omega\mapsto X(\omega)\mapsto e^{itX(\omega)}, $$ i.e. $$ \int h_t\circ X\, d\mathbb{P}=\int e^{itX}\, d\mathbb{P}=\mathbb{E}(e^{itX}). $$
(1) The problem lies in the definition of $e^{itx}$ when $x=\pm \infty$, as $\lim_{x\to +\infty} e^{itx}$ is not well defined. We are not bothered by this when $X\in\mathbb R$ almost surely, because we can define $\widetilde X$ a real valued random variable equal to $X$ almost everywhere. In particular, it won't change the integral.
(2) More generally, if $g\colon\mathbb R\to\mathbb R$ is a Borel measurable bounded function and $X$ a random variable, $$\mathbb E[g(X)]=\int_{\mathbb R}g(x)\mathrm dP_X(x).$$