Solve functional integral equations

Question

I encountered one problem which is to find $F(\lambda)$ which satisfies $1 = \int_{-\pi}^\pi e^{i k \lambda} dF(\lambda), \forall k \in \mathbb{Z}$ and the answer is $F(\lambda) = 1$ if $\lambda \ge 0$ and 0 otherwise.

I would like to solve it using mathematica, but a naive approach such as RSolve[1 == Integrate[Exp[I k x], {F[x], -Pi, Pi}], F, x, Assumptions -> Integers[k]] doesn't seem to work. The error message says

Supplied equations are not difference equations of the given functions.

Any help will be highly appreciated!

Answer 1

If F[x_]:=HeavisideTheta[x] then $dF=\delta(x)dx$, hence $\int_{-\pi}^{\pi}e^{ikx}dF=1$. The code is

F = HeavisideTheta[x]; Integrate[Exp[I k x] D[F, x], {x, -Pi, Pi}, 
 Assumptions -> k > 0]