# Revista Matemática Iberoamericana

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**Volume 26, Issue 1, 2010, pp. 147–174**

**DOI: 10.4171/RMI/598**

Published online: 2010-04-30

Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces

Anders Björn^{[1]}, Jana Björn

^{[2]}and Mikko Parviainen

^{[3]}(1) Linköping University, Sweden

(2) Linköping University, Sweden

(3) Helsinki University of Technology, Finland

We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have $L^q$-Lebesgue points everywhere.

*Keywords: *$mathcal{A}$-harmonic, fundamental convergence theorem, Lebesgue point, metric space, Newtonian function, nonlinear, *p*-harmonic, quasicontinuous, Sobolev function, superharmonic, superminimizer, supersolution, weak upper gradient

Björn Anders, Björn Jana, Parviainen Mikko: Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces. *Rev. Mat. Iberoam.* 26 (2010), 147-174. doi: 10.4171/RMI/598