We investigate in this chapter the strongly supermedian functions and kernels. The notion of strongly supermedian function has been introduced by J.F. Mertens [Mer 73] in the frame of a right process; see also [Mer 74]. Later on P.A. Meyer showed that these functions may be characterized in terms of the excessive functions, and clarified some of Mertens’ results from the analytical point of view (cf. [Me 73a], [Me 73b] and [MeTr 73]). Section 4.1 presents the λ-supermedian functionals and their representations as λ-strongly supermedian functions. It is also obtained the λ-quasi Lindelöf property for the fine topology. Section 4.2 exposes the regular λ-strongly supermedian functions and the regular supermedian λ-quasi kernels. Section 4.3 is devoted to the study of the strongly supermedian functions and kernels. It is obtained the Mertens decomposition of a finite strongly supermedian function as a sum of an excessive function and a regular strongly supermedian one. Relevant results on the regular strongly supermedian functions are also presented: the fine carrier and the associated strongly supermedian kernels. In Section 4.4 it is proved the remarkable fact if η, ξ are two excessive measures, η ≪ ξ, then the Radon-Nikodym derivative dη/dξ has a ξ-fine version, i.e. a ξ-version which is finite and finely continuous outside a set that is ξ-polar and ρ-negligible, ρ ◦ U being the potential component of ξ. Section 4.5 interprets the strongly supermedian functions and their properties in the probabilistic frame of the right processes, in terms of the homogeneous random measures.