  A charge space $(X,\mathcal{A},\mu)$ is a generalisation of a measure space,
consisting of a sample space $X$, a field of subsets $\mathcal{A}$ and a
finitely additive measure $\mu$, also known as a charge. Key properties a
real-valued function on $X$ may possess include $T_1$-measurability and
integrability. These properties are generalisations of corresponding properties
of real-valued functions on a (countably additive) measure space. However,
these properties are less well studied than their measure-theoretic
counterparts.
  This paper describes new characterisations of $T_1$-measurability and
integrability in the case that the charge space is bounded, that is, $\mu(X) <
\infty$. These characterisations are convenient for analytic purposes; for
example, they facilitate simple proofs that $T_1$-measurability is equivalent
to conventional measurability and integrability is equivalent to Lebesgue
integrability, if $(X,\mathcal{A},\mu)$ is a complete measure space.
  Several additional contributions to the theory of bounded charges are also
presented. New characterisations of equality almost everywhere of two
real-valued functions on a bounded charge space are provided. Necessary and
sufficient conditions for the function space $L_1(X,\mathcal{A},\mu)$ to be a
Banach space are determined. Lastly, the concept of completion of a measure
space is generalised for charge spaces, and it is shown that under certain
conditions, completion of a charge space adds no new equivalence classes to the
quotient space $\mathcal{L}_p(X,\mathcal{A},\mu)$.
