Let $X$ be a non-empty set, $a \in X$ and $\delta_a$ be the Dirac measure:
$$\delta_a(A)= \begin{cases} 1 &\text{if $a \in A$}\\ 0 &\text{if $a \notin A$}\\ \end{cases}$$
Consider the measure space $(X, \mathscr{P}(X), \delta_a)$
$$\mathcal{L}^1(X, \delta_a) = \left\{ f : X \to \overline{\mathbb{R}} : \text{$f$ is $\delta_a$-measurable and $\int_X |f|\,\mathrm{d}\delta_a < \infty$} \right\}$$
Let's say I want to determine the above set explicitly. Since $\int_X |f| \,\mathrm{d}\delta_a = |f(a)|$, does it mean that $\mathcal{L}^1(X, \delta_a)$ is simply the space of functions that don't take infinite values at $a$ ?