Consider some positive and real-valued functions $p, \tilde p :\mathbb{R}^2\to [0,\infty)$ that are integrable and decay sufficiently fast at infinity. For any entire function $f: \mathbb{C} \to \mathbb{C}$ the following integral vanishes $$\int_{\mathbb{R}^2} dx dy f(x+iy) \left(p(x,y) - \tilde p (x,y)\right) = 0.$$
I want to prove $$\forall a \in \mathbb{R}\, \forall\epsilon>0: \int_D dx dy \left(p(x,y) - \tilde p (x,y)\right) = 0,$$ where $D=\{(x,y)\in \mathbb{R}^2: \, f(x+iy)\in [a,a+\epsilon]\}$ for any entire function $f$.
For context, I want to show that despite averages complex weight functions do not have a unique real presentation, in the sense that I can find a unique real and positive probability density with the same expectation values of all holomorphic functions, above integrals are still equal.