Bernoulli shifts with bases of equal entropy are isomorphic

We prove that if \begin{document}$ G $\end{document} is a countably infinite group and \begin{document}$ (L, \lambda) $\end{document} and \begin{document}$ (K, \kappa) $\end{document} are probability spaces having equal Shannon entropy, then the Bernoulli shifts \begin{document}$ G \curvearrowright (L^G, \lambda^G) $\end{document} and \begin{document}$ G \curvearrowright (K^G, \kappa^G) $\end{document} are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both \begin{document}$ \lambda $\end{document} and \begin{document}$ \kappa $\end{document} have at least \begin{document}$ 3 $\end{document} points in their support. We furthermore produce finitary isomorphisms in the case where both \begin{document}$ L $\end{document} and \begin{document}$ K $\end{document} are finite.