  The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral
of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the
Mahler measure of a multivariate polynomial is the limit of Mahler measures of
univariate polynomials. We prove the analogous result for different extensions
of Mahler measure such as generalized Mahler measure (integrating the maximum
of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving
products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure
(involving $\log^k|P|$).
