I have a fixed real number $t\neq 0$ and a function with complex values but defined in $\mathbb{R}$: $$\ell: \mathbb{R} \to \mathbb{C}, \quad \ell(x) = e^{itx} - 1$$ For each $n \in \mathbb{N}$, I have a measure $M_n$ defined on the borelians of $\mathbb{R}$, such that: $$M_n \Big( \left[-\delta,\delta\right] \Big) = k_n, \quad k_n \uparrow\infty\,\, (n \to \infty)$$ for some $\delta>0$ fixed.
I want to show that $$\int_{-\delta}^{\delta} \ell(x) M_n d(x)\to \infty \quad (n \to \infty)$$
I'm trying to limit this integral to a lower bound that converges to infinity. For this, I'm trying to use $k_n$, but I don't know how to do that.
Help!
Update:
In fact, $M_n(\mathbb{R}) \to \infty$ with $M_n$ being a Lévy measure:
$\forall \delta > 0,\,\, \exists \, C_\delta< \infty, \,\, M_n(\mathbb{R} \setminus B_{\delta} (0) ) \leq C_\delta,\,\, \forall n$ and
$\exists \, C_0 < \infty, \, \int_{ |x| \leq 1} x^2 M_n (dx)< C_0,\,\, \forall n.$.
Futhermore, $M_n(E) = \sum_{j=1}^n \mu_{nj}(E)$ where $\mu_{nj}$ is not a dirac measure. This is in the context of the generalization of the central limit theorem for array of independent random variable!
As stated this is false. (Perhaps some crucial hypothesis is missing). Let $M_n=n\delta_0$. The $M_n[-\delta,\delta]=n \to \infty$ but $\int l(x)dM_n(x)=0$ for all $n$.