  We prove the Bernoulli property for determinantal point processes on $
\mathbb{R}^d $ with translation-invariant kernels. For the determinantal point
processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli
property was proved by Lyons and Steif and Shirai and Takahashi. As its
continuum version, we prove an isomorphism between the translation-invariant
determinantal point processes on $ \mathbb{R}^d $ with translation-invariant
kernels and homogeneous Poisson point processes. For this purpose, we also
prove the Bernoulli property for the tree representations of the determinantal
point processes.
