Let $X \in L_1$ be a positive random variable on the probability space $([0,1], \mathcal B, P)$, where $\mathcal B$ is the Borel $\sigma$ algebra on $[0,1]$. Consider
$$\phi(A) = E[X\mid A] \cdot I\big( P(A) \geq c \big),$$
for $A \in \mathcal B$ and $1 > c > 0$, and define $\phi^* = \sup\limits_{A \in \mathcal B} \phi(A)$. Here, $I(\cdot)$ is an indicator function.
Let $\{X_i\}_{i=1}^n$ be a random sample. Let $P_n:\mathcal B \rightarrow \mathbb R$ be the empirical measure; i.e. $P_n(\cdot)=\sum_{i=1}^n \delta_{X_i}(\cdot)$, where $\delta_{X_i}$ is the Dirac measure at $X_i$, and let $E_n[\cdot]=\int_{\Omega} \cdot dP_n $ be the empirical expectation. Consider
$$\phi_n(A) = \frac{E_n[X \cdot I(A)]}{P_n(A)} \cdot I\big( P_n(A) \geq c \big).$$
and define the estimator $\phi_n^* = \sup\limits_{A \in \mathcal B}\phi_n(A)$.
I am interested in what can be said about whether $\phi_n^*$ converges to $\phi^*$ almost surely (or in probability).
Observations:
For each fixed $A \in \mathcal B$ we have that $\phi_n(A) \longrightarrow \phi(A)$ almost surely by the strong law of large numbers.
For each $n$ we can find $A_n \in \mathcal B$ such that $\phi_n(A_n) = \phi_n^*$. I.e. $A_n$ is a candidate $A \in \mathcal B$ that maximizes $\frac{E_n[X \cdot I(A)]}{P_n(A)} \cdot I\big(P_n(A) \geq c \big)$.
If there is an $A^* \in \mathcal B$ such that $\phi^* = \phi(A^*)$, then we have that $$\phi_n(A_n) \geq \phi_n(A^*) \longrightarrow \phi(A^*) = \phi^*$$ almost surely, from obervation 1) and 2).
