  We define a motivic measure on the Berkovich analytification of an algebraic
variety defined over a trivially valued field, and introduce motivic
integration in this setting. The construction is geometric with a similar
spirit as Kontsevich's original definition, and leads to the formulation of a
functorial theory which mirrors, in this aspect, the approach of Cluckers and
Loeser via constructibe motivic functions. A version of the integral over
nontrivially valued fields and its relation to Hrushovski and Kazhdan's
integration are also discussed.
