$X[j]$ is $$\label{Mj}
M_{A,j} = \max_{x_j\in(x_{j,1},x_{j,2})} h_{A,j}(x_j).$$ The quality of $M_{A,j}$ in reducing the impurity of node $A$ is measured relative to the oracle impurity decrease at node $A$. Suppose we partition node $A$ into measurable sets $A_1$ and $A_2$ satisfying $\mathbb{P}_\alpha(A_1| A) = V_1,$ $A_1\cup A_2=\Omega,$ $A_1\cap A_2 = \emptyset,$ for all $X_1\in A_1,$ $X_2\in A_2,$ and $\mathbb{E}(Y|X_1)\le\mathbb{E}(Y|X_2),$ where $V_1\in[0,1]$ and $A_1, A_2$ are not required to be hyper rectangles. By definition, the set $A_1$ corresponds to the $V_1$ proportion of $A$ having the smallest $\mathbb{E}(Y|X)$ values, while the set $A_2$ corresponds to the $1-V_1$ proportion of $A$ having the largest $\mathbb{E}(Y|X)$ values. Similar to equation (\[p1alpha fs\]) and (\[delta Ialpha\]), we can define $p_{1,\alpha}, p_{2,\alpha}$ and the impurity decrease $\Delta I_\alpha(A,\mathbb{P})$. Given $V_1\in [0,1]$, the impurity decrease $\Delta I_\alpha(A,\mathbb{P})$ is unique, which is also the maximal impurity decrease for all measurable $A_1, A_2$ satisfying $\mathbb{P}_\alpha(A_1| A) = V_1,\; \mathbb{P}_\alpha(A_2|A) = 1-V_1$. Thus we can denote the impurity decrease as $\Delta I_\alpha(A,\mathbb{P}) = h^*_A(V_1)$. The oracle impurity decrease is the maximal value of $h^*_A(V_1)$: $M_{A}^* = \max_{V_1\in[0,1]} h^*_A(V_1).$ In general, the larger impurity decrease will tend to correspond to non-redundant features; this will particularly be the case under Conditions 