Sorry, I am an engineer and this surpasses completely my math education. If I am not precise, please comment on open/unclear assumptions, and I will do my best in correcting my mistakes.
QUESTION: is it possible to represent an integral $\lambda$ (functional with characteristic function $g(x)$) over the set of real numbers $\mathbb{R}$ with unknown/arbitrary real valued function $f(x)$
\begin{equation} \lambda = \int_{\mathbb{R}} g(x) f(x) dx \ , \end{equation}
and not specified but normalized, non-negative weight function $g(x)$ (not necessarily continuous)
\begin{equation} \int_{\mathbb{R}} g(x) dx = 1 \ , \qquad g(x) \geq 0 \quad \forall x \ , \end{equation}
in terms of Dirac distributions as follows?
\begin{equation} \lambda = \int_{\mathbb{R}} g(x) f(x) dx = \int_{\mathbb{R}} \sum_{i=1}^\infty c_i \delta(x_i - x) f(x) dx = \sum_{i=1}^\infty c_i f(x_i) \end{equation}
If yes, then is the characteristic function $g(x)$ (not necessarily constinuous) representable as
\begin{equation} g(x) = \sum_{i=1}^\infty c_i \delta(x_i - x) \ , \quad \sum_{i=1}^\infty c_i = 1 \ , \quad c_i \geq 0 \quad \forall i \end{equation}
for the functional $\lambda$?
Any recommendations for literature (standard books) for this would also be helpful. Thanks!
EDIT 1: Can this be interpreted in the sense of Riemann-Stieltjes integrals and corresponding convergent series? I just thought of this with $dG = g dx$
\begin{eqnarray} \lambda &=& \int_{\mathbb{R}} f(x) g(x) dx \\ &=& \int_{\mathbb{R}} f dG \\ &=& \sum_{i=1}^\infty f(x_i) (G(x_{i+1}) - G(x_i)) = \sum_{i=1}^\infty f(x_i) \Delta G_i \\ &=& \sum_{i=1}^\infty \int_{\mathbb{R}} f(x) \delta(x_i - x) dx \Delta G_i = \int_{\mathbb{R}} f(x) \left[ \sum_{i=1}^\infty \Delta G_i \delta(x_i - x) \right] dx \end{eqnarray}
So I dont know if this is correct
\begin{equation} g(x) = \sum_{i=1}^\infty \Delta G_i \delta(x_i - x) \end{equation}