Expression for characteristic function of a truncated RV Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function.  
Then if $A \in \mathscr{F}$ and $\mathbb{P}(A)>0$ then is there a simple expression for the characteristic function of the (truncated) random variable $\frac{X1_A}{\mathbb{P}(A)}$?

We denote by $\phi$ the characteristic function of the random variable $X1_A/\mathbb{P}(A)$. Then we have 
$$\phi(t)=\mathbb E\left(\mathbb 1_A\exp\left(it\frac X{\mathbb{P}(A)}  \right)\right)+1-\mathbb{P}(A).$$
I'm not sure that this can be simplified more or written as a function of the characteristic function of $X$ and the indicator function of $A$. 
If $A=\{X\in B\}$ where $B$ is a Borel subset of the real line, then 
$$\phi(t)= 1-\mathbb{P}(A)+\int_B e^{itx /\mathbb{P}(A)} f(x)\mathrm dx.$$ 
