$$
\begin{array}{l} \mathbb {E} \left[ \Psi^ {\mathcal {B}} (\widehat {\mathfrak {D}}) \right] = \underset {\widehat {\mathfrak {D}}} {\mathbb {E}} \left[ \sup _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \left(\mathcal {R} _ {\mathcal {D} _ {i}} ^ {\mathcal {B}} (\ell , h) - \mathcal {R} _ {\widehat {\mathcal {D}} _ {i}} ^ {\mathcal {B}} (\ell , h)\right) \right] \\ = \underset {\widehat {\mathfrak {D}}} {\mathbb {E}} \left\{\sup  _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \left[ \underset {(\tilde {\boldsymbol {x}} _ {i}, \tilde {y} _ {i}) \sim \mathcal {D} _ {i}} {\mathbb {E}} \left(\sup  _ {\tilde {\boldsymbol {x}} _ {i} ^ {\prime} \in \mathcal {B} (\tilde {\boldsymbol {x}} _ {i})} \ell (h (\tilde {\boldsymbol {x}} _ {i} ^ {\prime}), \tilde {y} _ {i})\right) - \frac {1}{n} \sum_ {j = 1} ^ {n} \sup  _ {\boldsymbol {x} _ {i j} ^ {\prime} \in \mathcal {B} (\boldsymbol {x} _ {i j})} \ell (h (\boldsymbol {x} _ {i j} ^ {\prime}), y _ {i j}) \right] \right\} \\ = \mathbb {E} _ {\widehat {\mathfrak {D}}} \left\{\sup  _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \frac {1}{n} \sum_ {j = 1} ^ {n} \left[ \underset {(\tilde {\boldsymbol {x}} _ {i j}, \tilde {\boldsymbol {y}} _ {i j}) \sim \widehat {\mathcal {S}} _ {i}} {\mathbb {E}} \left(\sup  _ {\tilde {\boldsymbol {x}} _ {i j} ^ {\prime} \in \mathcal {B} (\tilde {\boldsymbol {x}} _ {i j})} \ell (h (\tilde {\boldsymbol {x}} _ {i j} ^ {\prime}), \tilde {\boldsymbol {y}} _ {i j}) - \sup  _ {\boldsymbol {x} _ {i j} ^ {\prime} \in \mathcal {B} (\boldsymbol {x} _ {i j})} \ell (h (\boldsymbol {x} _ {i j} ^ {\prime}), y _ {i j})\right) \right] \right\} \\ \overset {a} {\leq} \underset {\widehat {\mathfrak {D}}, \widehat {\mathfrak {S}}} {\mathbb {E}} \left\{\sup  _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \frac {1}{n} \sum_ {j = 1} ^ {n} \left[ \sup  _ {\tilde {\boldsymbol {x}} _ {i j} ^ {\prime} \in \mathcal {B} (\tilde {\boldsymbol {x}} _ {i j})} \ell (h (\tilde {\boldsymbol {x}} _ {i j} ^ {\prime}), \tilde {y} _ {i j}) - \sup  _ {\boldsymbol {x} _ {i j} ^ {\prime} \in \mathcal {B} (\boldsymbol {x} _ {i j})} \ell (h (\boldsymbol {x} _ {i j} ^ {\prime}), y _ {i j}) \right] \right\} \\ \stackrel {b} {\leq} \underset {\widehat {\mathfrak {D}}, \widehat {\mathfrak {S}} ^ {\boldsymbol {\Sigma}}} {\mathbb {E}} \left\{\sup  _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \frac {1}{n} \sum_ {j = 1} ^ {n} \left[ \Sigma_ {i j} \left(\sup  _ {\tilde {\boldsymbol {x}} _ {i j} ^ {\prime} \in \mathcal {B} (\tilde {\boldsymbol {x}} _ {i j})} \ell (h (\tilde {\boldsymbol {x}} _ {i j} ^ {\prime}), \tilde {y} _ {i j}) - \sup  _ {\boldsymbol {x} _ {i j} ^ {\prime} \in \mathcal {B} (\boldsymbol {x} _ {i j})} \ell (h (\boldsymbol {x} _ {i j} ^ {\prime}), y _ {i j})\right) \right] \right\} \\ \stackrel {c} {\leq} 2 \underset {\widehat {\mathfrak {D}}, \boldsymbol {\Sigma}} {\mathbb {E}} \left\{\sup  _ {h \in \mathcal {H}} \frac {1}{t} \sum_ {i = 1} ^ {t} \frac {1}{n} \sum_ {j = 1} ^ {n} \left[ \Sigma_ {i j} \sup  _ {\boldsymbol {x} _ {i j} ^ {\prime} \in \mathcal {B} (\boldsymbol {x} _ {i j})} \ell (h (\boldsymbol {x} _ {i j} ^ {\prime}), y _ {i j}) \right] \right\} \\ \stackrel {d} {=} 2 \underset {\widetilde {\mathfrak {D}}} {\mathbb {E}} \left[ \Re_ {t n} (\widetilde {\mathcal {G}}) \right], \\ \end{array}
$$