\in \mathcal{F}_{j} \}$, $j \in \{1,\ldots, N\}$ (i.e the vector image of the evaluation of function $f \in \mathcal{{F}}_{j}$ on the sample $\mbs$). From the previous result, we have the next corollary (see also [@maurer2014inequality]): Now, assume we have a finite family of functional classes $\mathcal{F},\mathcal{F}_{1},\ldots,\mathcal{F}_{N}$, such that $\mathcal{F} = \bigcup_{i=1}^{N} \mathcal{F}_{i}$ and a sample $\mathbb{S}$ as before. Denote $A_{j} \subset \mathbb{R}^{n}, A_{j} = \{\paren{f(\bx_{1}),\ldots,f(\bx_{n})}: f \in \mathcal{F}_{j} \}$, $j \in \{1,\ldots, N\}$ (i.e the vector image of the evaluation of function $f \in \mathcal{{F}}_{j}$ on the sample $\mbs$). From the previous result, we have the next corollary (see also [@maurer2014inequality]):

\[cor:rad\_union\] For the empirical Rademacher complexity of the class $\mathcal{F}= \bigcup_{i=1,\ldots,N} \mathcal{F}_{i}$ based on the sample $\mathbb{S}$, we have: $$\begin{aligned}
    \hat{\mathfrak{R}}_{\mathbb{S}}\paren{\mathcal{F}} \leq \max_{m=1}^{N}\hat{\mathfrak{R}}_{\mathbb{S}}(\mathcal{F}_{m}) + 4\mathcal{M}\sqrt{\frac{\log{N}}{n}},
    \end{aligned}$$ where $\mathcal{M} = \sqrt{\sup_{f \in \mathcal{F}}\frac{1}{n}\sum_{i=1}^{n}f^{2}(\bx_{i})}$.

For the analysis of the class of regression decision trees algorithms , this means that we can in principle first analyze the Rademacher complexity of the predictor class for any fixed tree structure and splits, and then pay as a price the $\log$-cardinality of the union appearing in . One issue is that (because of the real-valued thresholds $t_s$) this union is not finite; however using a 