  The Ces\`aro limit - the asymptotic average of a sequence of real numbers -
is an operator of fundamental importance in probability, statistics and
mathematical analysis, including in many contexts in which sequences may be
unbounded. The Ces\`aro limit has many of the properties of an expectation
operator, and it is therefore natural to investigate conditions under which a
space of real-valued functions on the natural numbers may have a true
expectation operator in the form of a Ces\`aro limit. This paper introduces
pseudometric function spaces with this property, denoted $H_p(\calA)$ spaces.
These function spaces are shown to have quotients (denoted $\calH_p(\calA)$)
that are isometrically isomorphic to $\calL_p(\N,\calA,\nu)$ function spaces,
where $\calA$ is a field of sets (not necessarily a $\sigma$-field) and $\nu$
is a finitely additive measure, also known as a charge. The complete
$\calL_p(\N,\calA,\nu)$ spaces (and by implication, the $\calH_p(\calA)$ spaces
isomorphic to them) are characterised, and a sufficient condition for these
spaces to be separable is identified. In the process, substantial contributions
to the general theory of bounded charges are made, including convenient new
characterisations of $T_1$-measurability, integrability and equality almost
everywhere of functions on a general charge space $(X,\calA,\mu)$. A new
characterisation of completeness of $L_p$ spaces in this general context is
also presented.
