This part of the argument really is much more generally applicable; and it is part of Hutchinson's work on IFSs. I am currently teaching a class that is related, and I'll include the links to lecture recordings with time stamps below also.
To all that follows below one can refer to Hutchinson's paper "Fractals and Self-Similarity" (a retyped version of which is available at https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf) or Barnsley's book Fractals Everywhere.
To fix the notations, let $X$ be a complete metric space, $\mathcal{W}=\{w_1,w_2,...,w_N\}\subseteq F(X;X)$ be a finite collection of functions from $X$ to itself such that
$$
\forall i: \operatorname{Lip}(w_i)\in[0,1[
$$
so that each $w_i$ is a contraction (see e.g. Question on the distortion of a metric embedding and Lipschitz maps). I'll call any $\mathcal{W}\curvearrowright X$ like above a B-IFS ("iterated function system in the sense of Barnsley").
Hutchinson Attractor Theorem: For any B-IFS $\mathcal{W}\curvearrowright X$, there is a unique nonempty compact subset $A_\mathcal{W}\subseteq X$, called the attractor of the B-IFS, that solves the set equation
$$
A_\mathcal{W}=\bigcup_{i=1}^N w_i(A_\mathcal{W}).
$$
(See Theorem 9.1 on p.124 in Falconer for $X=\mathbb{R}^d$ (the proof works for a general metric space $X$), or alternatively https://youtu.be/XydQoxKnodE?feature=shared&t=7752)
Denote next by $\Sigma_N^+$ the set of all sequences $\omega_\bullet:\mathbb{Z}_{\geq0}\to \{1,2,...,N\}$, $\sigma:\Sigma_N^+\to\Sigma_N^+$ be the shift transformation defined by $\sigma(\omega_\bullet)=\omega_{\bullet+1}$, that is
$$
\sigma(\omega_0,\omega_1,...,\omega_n,...)=(\omega_1,\omega_2,...,\omega_n,...).
$$
Fix a $\theta\in\mathbb{R}_{>1}$ and endow the shift space $\Sigma_N^+$ with the metric $d_\theta$ defined by
$$
d_\theta(\omega,\eta)=\sum_{n\geq 0}\dfrac{\delta(\omega_n,\eta_n)}{\theta^n},
$$
where $\delta(\omega_n,\eta_n)$ is $1$ if $\omega_n\neq\eta_n$ and $0$ otherwise. Note that $\sigma$ is Lipschitz relative to $d_\theta$, and that any two $d_\theta$ metrics are Hölder equivalent.
Define the function
$$
\varphi_\mathcal{W}:\Sigma_N^+\times\mathbb{Z}_{\geq0}\to F(X;X), \varphi_\mathcal{W}(\omega,n)=w_{\omega_0}\circ w_{\omega_1}\circ \cdots \circ w_{\omega_{n-1}}.
$$
Then $\varphi_\mathcal{W}$ is a cocycle over $\sigma$ (see e.g. Intuition of cocycles and their use in dynamical systems or https://youtu.be/ei-RND8EmQc?feature=shared&t=7973), that is
$$
\varphi_{\mathcal{W}}(\omega,n+m)=\varphi_{\mathcal{W}}(\omega,n)\circ \varphi_{\mathcal{W}}(\sigma^n(\omega),m).
$$
Hutchinson Coding Theorem: Let $\mathcal{W}=\{w_1,w_2,...,w_N\}\curvearrowright X$ be a B-IFS. Then
- $\forall x\in X, \forall \omega\in\Sigma_N^+: \Phi_\mathcal{W}(\omega)=\lim_{n\to\infty}\varphi_\mathcal{W}(\omega,n)(x)\in A_\mathcal{W}$ exists and is independent of $x$.
- $\Phi_\mathcal{W}:\Sigma_N^+\to A_\mathcal{W}$ is Hölder continuous and onto.
(See e.g. https://youtu.be/K1eGO1iIBqc?feature=shared&t=1193)
Recall that if $\Psi:M\to N$ is a continuous function between metric spaces, then for any Borel probability measure $\mu$ on $M$, its pushforward $\nu=\Psi_\ast(\mu)$ under $\Psi$ defined by $\Psi_\ast(\mu)(B)=\mu(\Psi^{-1}(B))$ is a Borel probability measure on $N$ (see e.g. Constructing the Haar measure of the $n$-dimensional Torus or Do full rank matrices in $\mathbb Z^{d\times d}$ preserve integrals of functions on the torus?).
Let $\mathcal{W}=\{w_1,w_2,...,w_N\}\curvearrowright X$ be a B-IFS with $N$ contractions; fix a probability vector $p=(p_1,p_2,...,p_N)\in\mathbb{R}^N$ (so that each $p_i$ is a nonnegative number and $\sum_i p_i=1$). Then we have a probability measure $\xi_p$ on $\mathcal{W}$ defined by
$$
\xi_p(\{w_i\})=p_i,
$$
so that for any subset $\mathcal{W}'\subseteq \mathcal{W}$,
$$
\xi_p(\mathcal{W}')=\sum\{p_i| w_i\in \mathcal{W}'\}.
$$
The choice of $p$ also defines a unique $\sigma$-invariant Borel probability measure $\mu_p$ on $\Sigma_N^+$ (see e.g. Are product measures the only $T$-invariant measures if $T$ is the left shift on $\{0,1,\ldots,k-1\}^\mathbb Z$?) defined by
\begin{align*}
&\forall n\in\mathbb{Z}_{\geq0},\forall \iota_0,\iota_1,...,\iota_{n-1}\in\{1,2,...,N\}:\\
&\mu_p(\{\omega\in\Sigma_N^+| \forall i<n: \omega_i=\iota_i\}) = p_{\iota_0}p_{\iota_1}...p_{\iota_{n-1}}.
\end{align*}
Then the pushforward $\nu_{\mathcal{W},p}=(\Phi_\mathcal{W})_\ast(\mu_p)$ is a Borel probability measure on the attractor $A_\mathcal{W}\subseteq X$.
For $M$ a metric space; denote by $\operatorname{ev}:C^0(M;M)\times M\to M$ the evaluation map that is defined by $\operatorname{ev}(f,x)=f(x)$. We may think of $\operatorname{ev}$ as a dynamical system (see e.g. How a group represents the passage of time?). Here $C^0(M;M)$ is the monoid of all continuous self-maps of $M$. Note that $C^0(M;M)$ is in and of itself a metric space with the supremum norm. Let $\xi$ be a Borel probability measure on $C^0(M;M)$. It is sometimes instructive to consider $\xi$ to be the timespace of some dynamical system (a "varimonoid" or "varigroup"; see e.g. The meaning of Blow-up of a measure in a point). A Borel probability measure $\nu$ on $M$ then is called $\xi$-stationary if
$$\operatorname{ev}_\ast(\xi\otimes \nu)=\nu.$$
(Here $\otimes$ signifies the product measure; see e.g. Convolution of discrete measures)
More explicitly this means, for any Borel measurable $B\subseteq X$:
$$
\nu(B)=\int_{\operatorname{supp}(\xi)} f_\ast(\nu)(B) \,d\xi(f).
$$
(Here $\operatorname{supp}(\xi)$ is the support of $\xi$, that is, the smallest closed subset of $C^0(M;M)$ of full $\xi$-measure; see e.g. What can we say about the pre-image of the support of the pushforward measure in relation to some original measure? .)
Let $\mathcal{W}=\{w_1,w_2,...,w_N\}\curvearrowright X$ be a B-IFS with $N$ contractions; fix a probability vector $p=(p_1,p_2,...,p_N)\in\mathbb{R}^N$. Then one can rewrite the condition for a Borel probability measure $\nu$ on $X$ to be $\xi_p$-stationary by asking for
$$
\nu(B)=\sum_{i=1}^N p_i \,(w_i)_\ast(\nu)(B).
$$
to be satisfied for any Borel measurable $B\subseteq X$.
Hutchinson Measure Theorem: For any B-IFS $\mathcal{W}=\{w_1,w_2,...,w_N\}\curvearrowright X$ and for any probability vector $p=(p_1,p_2,...,p_N)$, the measure $\nu_{\mathcal{W},p}$ supported on the compact subset $A_\mathcal{W}\subseteq X$ defined above is the unique $\xi_p$-stationary Borel probability measure whose support is bounded.
To tie all this to the Moran-Hutchinson dimension formula whose proof you are studying, for $\mathcal{W}=\{w_1,w_2,...,w_N\}\curvearrowright X$ a B-IFS let $s\in\mathbb{R}_{\geq0}$ be the unique number such that
$$
\sum_{i=1}^N\operatorname{Lip}(w_i)^s=1.
$$
Let's call $s$ the upper similarity dimension of the B-IFS $\mathcal{W}\curvearrowright X$.
(We put $s=0$ if all $w_i$'s are constant functions.)
Then by definition $q^s=(\operatorname{Lip}(w_1)^s,\operatorname{Lip}(w_2)^s,...,\operatorname{Lip}(w_N)^s)$ is a probability vector (if all $w_i$'s are constant functions we may define $q^s=(1/N,1/N,...,1/N)$). Applying the above construction we have a measure $\nu_{\mathcal{W},q^s}$ on the attractor $A_\mathcal{W}$; and this measure is exactly your measure $\widetilde{\mu}$. Thus $\widetilde{\mu}=\nu_{\mathcal{W},q^s}$ can be considered as a dynamically significant measure on the attractor $A_\mathcal{W}$. The upper similarity dimension $s$ also is a candidate for the Hausdorff dimension of $A_\mathcal{W}$, and part of the Moran-Hutchinson argument is that, under the assumptions of
- $X$ is a Euclidean space,
- Each $w_i$ is a similitude,
- $\mathcal{W}\curvearrowright X$ satisfies OSC,
the dynamically significant measure $\nu_{\mathcal{W},q^s}$ and the geometrically significant $s$-Hausdorff measure $\mathfrak{H}^s$ are intimately connected to each other. (Falconer does not seem to mention this, but in fact in this case one has $\nu_{\mathcal{W},q^s}=\mathfrak{H}^s(\bullet|A_\mathcal{W})$; that is, the dynamically significant measure is the $s$-Hausdorff measure conditioned on the attractor.)
(In class we just proved $\mathfrak{H}^s(A_\mathcal{W})<\infty$ and started the proof of $\mathfrak{H}^s(A_\mathcal{W})>0$, and we haven't covered stationary measures yet.)
