Basic workings of probability theory on $\mathcal{C}([0,1])$ I am confused as to how exactly the underlying probablity space $(\Omega,\mathscr{F},\mathbb{P})$ relates to a probability measure $\mu:\mathcal{B}(\mathcal{C})\longrightarrow[0,1]$ where $\mathcal{C}:=\mathcal{C}([0,1])$ is the class of continuous functions $f:[0,1]\longrightarrow\mathbb{R}$ and $\mathcal{B}(\mathcal{C})$ is the Borel $\sigma$-field of subsets of $\mathcal{C}$.
As per the excellent question and answer here Natural & important probability measures on $\mathcal{C}[0,1]$, in particular the Wiener measure, I have a natural preference to avoid, at least initially, the stochastic process viewpoint. However in case this is relevant to my questions, my understanding is as follows: for each $f\in\mathcal{C}$ and for fixed $t\in[0,1]$, $f_{t}(\omega):\Omega\longrightarrow\mathbb{R}$ is a random variable as $\omega$ runs through $\Omega$ if $f$ is $\mathscr{F}/\mathscr{R}$ measurable ($\mathscr{R}$ being the linear Borel sets). I think each $f$ are measurable since continuity implies measurability. On the other hand for fixed $\omega\in \Omega$, $f_{t}(\omega):[0,1]\longrightarrow\mathbb{R}$ is a path as $t$ runs through $[0,1]$, i.e. a realisation of a real valued function on $[0,1]$. Thus using the dot notation to signify a variable, $f_{t}(\cdot)$ is a random variable at $t$ and $f_{(\cdot)}(\omega)$ is the entire path of $f$ at $\omega$. Furthermore $(f_{t}(\omega))_{\omega,t}:=(f_{t}(\omega))_{\omega\in\Omega,t\in[0,1]}$ is a stochastic process $f:\Omega\times[0,1]\longrightarrow\mathbb{R}$.
Random elements in $\mathcal{C}$
I now, perhaps naively, apply my knowledge of how the basics of probability theory work for $X$ - i.e. a random variable in a standard probability set-up)
Suppose $\tilde{F}:\Omega\longrightarrow\mathcal{C}$ is a $\mathscr{F}/\mathcal{C}$-measurable mapping between the measurbale spaces $(\Omega,\mathscr{F})$ and $(\mathcal{C},\mathcal{B}(\mathcal{C}))$, i.e. for each $\omega\in\Omega$, $\tilde{F}(\omega)\mapsto \tilde{f}$ for some $\tilde{f}\in\mathcal{C}$. Then $\tilde{F}$ is a random element in $\mathcal{C}$ and is analagous to a $\mathscr{F}/\mathscr{R}$-measurable mapping $X(\omega)\mapsto x$ for some $x\in\mathbb{R}$ - i.e. a standard random variable. According to Billinglsey (Convergence of Probability Measures, 2nd Ed, 1999) elements can mean scalars, vectors, sequences as well, and I am trying to follow this general language so I understand probability theory applied to metric spaces (of which $(\mathcal{C},d_{\infty})$ where $d_{\infty}$ is the sup metric is the most general I wll probably ever have the need to consider). The open $d_{\infty}$-balls $B_{\epsilon}(f)=\{g\in\mathcal{C}:d_{\infty}(f,g)<\epsilon\}$ centred at $f$ of radius $\epsilon$ are conceptually the same as in Euclidean space: i.e. the class of $d_{\infty}$-open sets of $\mathcal{C}$ denoted $\mathcal{U}(\mathcal{C})$ satisfy each $U\in \mathcal{U}(\mathcal{C})$ being a union of open balls, and the metric space $(\mathcal{C},d_{\infty})$ is separable and complete. The Borel $\sigma$-field $\mathcal{C}$ is crucially also generated by $\mathcal{U}(\mathcal{C})$ (this is crucial in the sense that oftentimes proving a result on the generating class then extends to the entire $\sigma$-field). However here I do not need to use such properties here.
Let $\require{enclose}
     \enclose{horizontalstrike}{\mu_{\tilde{F}}:\mathcal{C}\longrightarrow[0,1]}$ $\mu_{\tilde{F}}:\mathcal{B}(\mathcal{C})\longrightarrow[0,1]$
be the probability law of $\tilde{F}$: what does this mean and how does $\mu_{\tilde{F}}$ assign probabilities to elements in $\mathcal{C}$?;
$$\mu_{\tilde{F}}(B)=\mathbb{P}\left\{\omega\in\Omega:\tilde{F}(\omega)\in B\right\}\hspace{20pt}\text{ for all }B\in\mathcal{B}(\mathcal{C})$$
In words: $B$ is a set of continous real-valued functions on $[0,1]$ and $\tilde{F}(\omega)\in B$ means the entire path $\tilde{F}(\omega):=\tilde{f}_{(\cdot)}(\omega)$ at $\omega$ lies in $B$. The above often gets shortened to $\mu_{\tilde{F}}(B)=\mathbb{P}\{\tilde{F}\in B\}=\mathbb{P}\{\tilde{F}^{-1}(B)\}$.
Measurable mappings and intergals of Measurable mappings of random elements in $\mathcal{C}$
If my understanding about random elements is OK, how are measurable mappings of these random elements defined, and furthermore how are intergals of these mappings of elements defined? Once again I now perhaps naively apply my knowledge of how this works for $X$). Let $\phi$ be a positive-valued $\mathcal{C}/\mathscr{R}$-measurable function (i.e. function of functions such as $\phi(\tilde{f})\mapsto\text{max}\left\{|\tilde{f}_{t}|:t\in[0,1]\right\}$). Then $\phi(\tilde{F})=\phi\circ \tilde{F}$ is a $\mathscr{F}/\mathscr{R}$-measurable mapping (theorem 13.1 (ii), Billingsley 1995, Proabability and Measure, 3rd Ed), and as far as I can tell, I am free now to apply standard measure theory pertaining to the integral of random variables to this mapping - i.e. no new theory of integration required (Billingsley 1995 uses a quite general set-up so perhaps here I am benefiting from investing in this extra generality).
Using the defintion of expectation of random variables, the change of variable theorem (lines 5 and 6), and the definition of the integral for positive mappings;
$$\begin{align*}
\require{enclose}
     \enclose{horizontalstrike}{\mathbb{E}_{\mu_{\tilde{F}}}\left[\phi(\tilde{F})\right]
}\mathbb{E}_{\mu_{\tilde{F}}}\left[\phi\right]&:=\int_{\mathcal{C}}\phi(g)\mu_{\tilde{F}}(dg)\\
&:=\text{sup}\sum_{i}\left\{\left[\underset{g\in B_{i}}{\text{inf}}\phi(g)\right]\mu_{\tilde{F}}(B_{i})\right\}\\
&=\text{sup}\sum_{i}\left\{\left[\underset{g\in B_{i}}{\text{inf}}\phi(g)\right]\mathbb{P}\left[\omega:\tilde{F}(\omega)\in B_{i}\right]\right\}\\
&=\text{sup}\sum_{i}\left\{\left[\underset{g\in B_{i}}{\text{inf}}\phi(g)\right]\mathbb{P}\left[\tilde{F}^{-1}(B_{i})\right]\right\}\\
&=\int_{\mathcal{C}}\phi(g)\mathbb{P}\circ\tilde{F}^{-1}(dg)\\
&=\int_{\Omega}\phi(\tilde{F}(\omega))\mathbb{P}(d\omega)\\
&:=\text{sup}\sum_{i}\left\{\left[\underset{\omega\in F_{i}}{\text{inf}}\phi(\tilde{F}(\omega))\right]\mathbb{P}(F_{i})\right\}\\
&:=\mathbb{E}_{\mathbb{P}}\left[\phi(\tilde{F})\right],
\end{align*}$$
where the supremums extend over all finite decompositions $\{B_{i}\}$ of $\mathcal{C}$ into $\mathcal{B}(\mathcal{C})$-sets and all finite decompositions $\{F_{i}\}$ of $\Omega$ into $\mathscr{F}$-sets.
This all seems perfectly reasonable to me but please do point out where I have gone wrong.
 A: There is no mistake in your reasoning, but some notational imprecision which might be a source of confusion. Firstly:

Let $\mu_{\tilde{F}}:\mathcal{C}\longrightarrow[0,1]$ be the probability law of $\tilde{F}$

A measure is a functional on a $\sigma$-algebra, so one should write $\mu_{\tilde{F}}:\mathcal B(\mathcal C)\longrightarrow[0,1]$.
Secondly, whereas your understanding of what the law of a random variable is is completely correct, one might want give as a definition what you derive as a conclusion. (The two points of view are obviously and immediately equivalent, but somewhat different in spirit).
Say $(S,\mathcal S)$ is a measurable space, and $X$ is an $S$-valued random variable on a probability space $(\Omega,\mathscr F,\mathbb P)$. (That is, in analytical terms, $X$ is an $\mathscr F/\mathcal S$-measurable function.)

The law $\mu$ of $X$ is the measure $X_*\mu:= \mu \circ X^{-1}$

i.e. the push-forward measure of $\mu$ via $X$.
Now to the last bunch of formulas.

$$\mathbb{E}_{\mu_{\tilde{F}}}\left[\phi(\tilde{F})\right]:=\int_{\mathcal{C}}\phi(g)\mu_{\tilde{F}}(dg)$$

The left-hand side as you write it here does not make sense. Indeed, $\mu_{\tilde{F}}$ is a measure on $\mathcal C$, whereas $\phi(\tilde{F})$ (which here should really be written $\phi\circ \tilde F$, for clarity) is a function on $\Omega$. One cannot integrate a function defined on some measurable space w.r.t. a measure on another measurable space.
I assume (from the $:=$ symbol) that what was meant is
$$\mathbb{E}_{\mu_{\tilde{F}}}\left[\phi\right]:=\int_{\mathcal{C}}\phi(g)\mu_{\tilde{F}}(dg)$$
which is the definition of expectation.
Then,
$$
\mathbb{E}_{\mu_{\tilde{F}}}\left[\phi\right]=\mathbb{E}_{\mathbb{P}}\left[\phi(\tilde{F})\right],
$$
is nothing but the standard change of variable formula for push-forward measures, which you basically reproved (again correctly).
