Let $V$ be a complex vector space. By an $\infty$-sesquilinear form, I mean a sesquilinear form that may take valued in $\mathbb{C} \cup \infty$. If $X$ is a space then, in some sense, $\infty$-sesquilinear forms on are "generalizations of measures" since a measure $\mu$ induces a $\infty$-sesquilinear form on the complex valued Borel functions on $X$ via the usual formula $\langle f,g\rangle = \int_X f\overline{g}d\mu$. If one has a $\infty$-sesquilinear form on some complex vector space, then it is easy to define an $L^2$, via restricting to the subspace of functions that the form takes finite values, then by quotienting out by the vectors where the form vanishes, and finally taking the metric completion of the resulting inner product space.
Is there a natural way to define a version of $L^1$ that generalizes the case of measures just using the sesquilinear form? I do not personally have a feel for what can happen.
I appreciate any ideas or references!