This paper deals with aspects of measure and integration in the context of point-free topology. It establishes the Lebesgue integral for general functions on \( \sigma \)-locales, extending the author's study for simple functions in [5]. The integral is described with respect to a measure defined on the coframe of all \( \sigma \)-sublocales. This makes it possible to define the notion of integrable function for any function and to compute the integral over any \( \sigma \)-sublocale of L. In particular, it is shown how the new point-free integral generalises the classical Lebesgue integral.