Consider these sets $$ A\equiv \cap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ B_n\equiv \cap_{\delta>0} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ C_n(L)\equiv \cap_{0<\delta\leq L} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}=\{x \in X: d(p_n, [\ell(x), u(x)])\leq L\} $$ $$ C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])\leq L_n\} $$ where:

• $(p_n)_n$ is a sequence of reals taking values in $[0,1]$.
• $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
• $L_n$ is a sequence of strictly positive reals converging to zero as $n$ goes to infinity.

Could you help me to show that $$ d_H(A, B_n)\rightarrow_{a.s.} 0, $$ where $$ d_H(A, B_n)\equiv \max\{\sup_{x\in B_n}d(x,A), \sup_{x\in A}d(x, B_n)\}, $$ is the Hausdorff distance. Also, can we say something about $d_H(A, C_n(L))$ and $d_H(A, C_n(L_n))$?