genesis of the isomorphism $\Lambda(\Gamma) \cong \mathbb Z_p[[T]]$ and Mahler's theorem, etc.

Question

I am reading Coates and Sujatha's book "Cyclotomic fields and zeta values" where we see the following isomorphism in section 3.3. I write $\Gamma$ for $\mathbb Z_p$ just to have multiplicative notation for the group for convenience. $$\Lambda(\Gamma) \xrightarrow{M} \mathbb Z_p[[T]]$$ given by $$\lambda \in \Lambda(\Gamma) \rightsquigarrow \sum_{n \geq 0} c_n(\lambda)T^n,$$ where $c_n(\lambda)$ is defined to be $\int_{\mathbb Z_p}{x \choose n}d\lambda$. Later an inverse is produced using Mahler's theorem which asserts the possibility of writing $p$-adically continuous functions $f(x)$ as a series in $x \choose n$ with unique coefficients $a_n \to 0$. More precisely, if $g(T)=\sum_{n\geq 0}c_nT^n \in \mathbb Z_p[[T]]$, then the element $\lambda$ of $\Lambda(\Gamma)$ that is $M^{-1}(g)$ is determined by how it integrates continuous function $f(x)=\sum_{n \geq 0}a_n{x \choose n}$: $$\int_{\mathbb Z_p}f(x)d\lambda:=\sum_{n \geq 0}a_nc_n.$$ Equipped with these maps, one easily verifies that M is a $\mathbb Z_p$-algebra isomorphism. Although it is easy to verify that these are isomorphisms I don't get how they came up with these maps; it appears as though they are drawn out of thin air: I understand that elements of $\Lambda(\Gamma)$ can be identified with $\mathbb Z_p$-valued measures on $\Gamma (\cong \mathbb Z_p)$. Following this natural identification, how can one arrive at the maps $M$ and $M^{-1}$? I can see following the definition of integral and the well-known isomorphism $\mathbb Z_p[[\Gamma]] \cong \mathbb Z_p[[T]] \cong \varprojlim \mathbb Z_p[T]/((1+T)^{p^n}-1)$, that $$\int_{\mathbb Z_p} x^k d(M^{-1}((1+T)^m))=m^k,$$ and consequently $$\int_{\mathbb Z_p} {x \choose n} d(M^{-1}((1+T)^m))={m \choose n}.$$ But how this leads to the isomorphism $M$ above is not something I don't understand. Any insights are welcome. Thankyou very much!