Conditional Expectation of a mathematical function

Question

Suppose $(X,d)$ is a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff dimension of set $A$.

When defining $n\in\mathbb{N}$, where set $A\subseteq\mathbb{R}^{n}$ and function $f:A\to\mathbb{R}$; the expected value of $f$ is

$$\mathbb{E}[f]=\frac{1}{H^{\text{dim}_{\text{H}}(A)}(A)}\int_{A}f \; dH^{\text{dim}_{\text{H}}(A)}$$

which is undefined if $H^{\text{dim}_{\text{H}}(A)}(A)=0$ or $+\infty$ (e.g. $A=\mathbb{Q}$).

One way to get a defined value is to use conditional expectation; however, I don't understand the definition.

I was wondering how it compares to the following approach, where we define some sequence of sets and then an extension of $\mathbb{E}[f]$:

Definition 1 ($\star$-sequence of sets) Suppose we define a sequence of sets $(F_r^{\star})_{r\in\mathbb{N}}$, where:

1. The set theoretic limit of $(F_r^{\star})_{r\in\mathbb{N}}$ is $A$ (i.e., $(F_r^{\star})_{r\in\mathbb{N}}$ converges to $A$) where

$$\limsup_{r\to\infty}F_r^{\star}=\bigcap_{r\ge 1}\bigcup_{q\ge r}F_q^{\star}$$ $$\liminf_{r\to\infty}F_r^{\star}=\bigcup_{r\ge 1}\bigcap_{q\ge r}F_q^{\star}$$

$\quad\quad$the set-theoretic limit should be:

$$\limsup_{r\to\infty}F_r^{\star}=\liminf_{r\to\infty}F_r^{\star}=A$$

2. For all $r\in\mathbb{N}$, $0<H^{\text{dim}_{\text{H}}(F_r^{\star})}(F_r^{\star})<+\infty$

we then have $(F_r^{\star})$ is a $\star$-sequence of sets or starred-sequence of sets.

Definition 2 (Generalized Expected Value) If $(F_r^{\star})_{r\in\mathbb{N}}$ is a $\star$-sequence of sets (def. 1), the generalized expected value of $f$ w.r.t $(F_r^{\star})_{r\in\mathbb{N}}$ is $E^{**}[f,F_r^{\star}]$ where:

\begin{align} & \small{\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{{H}^{\text{dim}_{\text{H}}(F_r^{\star})}\left(F_r^{\star}\right)}\int_{F_r^{\star}}f\, d{H}^{\text{dim}_{\text{H}}(F_r^{\star})}-\mathbb{E}^{**}[f,F_r^{\star}]\right|< \epsilon\right)} \end{align}

Question: How does $E^{**}[f,F_r^{\star}]$ compare with conditional expectation? How do we define conditional expectation for function $f$?