Let $\mathcal{H}$ be ahypothesis class, $h\in \mathcal{H}$ be a function a model that maps an input space $\mathcal{X}$ to $\{0,1\}$, and $\epsilon > 0$, let $\mathcal{D}$ denotes the distribution which generates elements of $\mathcal{X}$ Let $ B(h,\epsilon)$ denote the ball centered in $h$ of radius $\epsilon$.
We denote $\mathcal{A}_h(S_q) = \hat{h}$ a learning algorithm that takes as input the query $S_q$ and output a hypothesis $h \in \mathcal{H}$: $\mathcal{A}_h(S_q) = \hat{h}$.
Let $S_q = \{X_1, \dots, X_q \}$ (random vector), We sat that $S_q \in \mathcal{Agg} \Big( B(h,\epsilon)\Big)$ if:
$$\forall i \in \{1, \dots, q\}, \forall f_1,f_2 \in B(h,\epsilon): f_1(X_i) = f_2(X_i)$$
Now consider the event: $$\mathcal{G}_h = \{S: ||\mathcal{A}(S_q) - h|| < \epsilon \}$$
I'm interested in the following expectation:
$$\mathbb{E} \Big[\mathbb{1}_{\mathcal{G}_h}(S_q)|S_q \in \mathcal{Agg} \Big( B(h,\epsilon)\Big) \Big]$$
Is this a zero quantity?
I am not sure if my argument is correct, but since the mass is concentrated on a region where infinite elements agree with $h$, then the algorithm should output a candidate hypothesis inside the ball $B(h,\epsilon)$.
