In this paper, a new nonlinear generalization of linear canonical correlation analysis (LCCA) is derived. This framework, called measure transformed canonical correlation analysis (MTCCA), applies LCCA to the considered pair of random vectors after transformation of their joint probability distribution. The proposed transform is structured by a pair of nonnegative functions called the MT-functions. It preserves statistical independence and maps the joint probability distribution into a set of joint probability measures on the joint observation space. Specification of MT-functions in the exponential family, leads to MTCCA, which, in contrast to LCCA, is capable of detecting nonlinear dependencies. In the paper, MTCCA is illustrated for recovery of a nonlinear system with known structure, and for construction of networks that analyze long-term associations between companies traded in the NASDAQ and NYSE stock markets.