  We propose a novel ensemble method called Riemann-Lebesgue Forest (RLF) for
regression. The core idea in RLF is to mimic the way how a measurable function
can be approximated by partitioning its range into a few intervals. With this
idea in mind, we develop a new tree learner named Riemann-Lebesgue Tree (RLT)
which has a chance to perform Lebesgue type cutting,i.e splitting the node from
response $Y$ at certain non-terminal nodes. We show that the optimal Lebesgue
type cutting results in larger variance reduction in response $Y$ than ordinary
CART \cite{Breiman1984ClassificationAR} cutting (an analogue of Riemann
partition). Such property is beneficial to the ensemble part of RLF. We also
generalize the asymptotic normality of RLF under different parameter settings.
Two one-dimensional examples are provided to illustrate the flexibility of RLF.
The competitive performance of RLF against original random forest
\cite{Breiman2001RandomF} is demonstrated by experiments in simulation data and
real world datasets.
