This paper investigates a broad class of non-Gaussian measures, $ μ_Ψ$, associated with a family of generalized Wright functions, $_mΨ_q$. First, we study these measures in Euclidean spaces $\mathbb{R}^d$, then define them in an abstract nuclear triple $\mathcal{N}\subset\mathcal{H}\subset\mathcal{N}'$. We study analyticity, invariance properties, and ergodicity under a particular group of automorphisms. Then we show the existence of an Appell system which allows the extension of the non-Gaussian Hilbert space $L^2(μ_Ψ)$ to the nuclear triple consisting of test functions' and distributions' spaces, $(\mathcal{N})^{1}\subset L^2(μ_Ψ)\subset(\mathcal{N})_{μ_Ψ}^{-1}$. Furthermore, thanks to the definition of two transformations, $S_{μ_Ψ}$ and $T_{μ_Ψ}$, we study Donsker's delta as an element within $(\mathcal{N})_{μ_Ψ}^{-1}$ applying the integral equations fulfilled by $_mΨ_q$.