Scholium

For every $\dot{f} \in L^1_{\mathrm{loc}}(\mathrm{T}, \mu; \mathbf{R})$, set $\varphi(\dot{f}) = f \cdot \mu$, where $f \in \dot{f}$; the mapping $\varphi$ is linear, increasing and injective (Cor. 2 of Prop. 3), and its image in $\mathscr{M}(T)$ is the band B generated by $\mu$ (Cor. 2 of Th. 2). The mapping $\varphi$ therefore permits identifying $L^1_{\text{loc}}(T, \mu; \mathbf{R})$ with a space of real measures on T; since all of the spaces $L^p_R(T, \mu)$ are subspaces of $L^1_{\text{loc}}(T, \mu; \mathbf{R})$, they too may be identified with subspaces of $\mathscr{M}(T)$. Analogous considerations hold for complex-valued functions and measures. Note that the mapping $\varphi$ considered above is an isomorphism of the ordered vector space structures of $L^1_{\text{loc}}$ and B, but is obviously not an isomorphism for the topological vector space structures of these spaces.

Since every band in a fully lattice-ordered space is itself a fully lattice-ordered space (Ch. II, §1, No. 5), one sees that the space $L^1_{\text{loc}}$ is fully lattice-ordered; but it is worthwhile to recall that the supremum in $L^1_{\text{loc}}$ of an uncountable family $(\dot{f}_\alpha)$ of equivalence classes is not necessarily identical to the class of the upper envelope of the functions $f_\alpha$. However, we saw that for an increasing sequence $(f_n)$ of locally $\mu$-integrable functions whose upper envelope $f$ is locally $\mu$-integrable, $f \cdot \mu$ is the supremum of the sequence of measures $(f_n \cdot \mu)$ in $\mathscr{M}(T)$ (Cor. of Prop. 6).

Here is an interesting consequence of Corollary 3 of Th. 2: