Proposition 3

— Let $\nu$ be a measure on T. For $\nu$ to be of the form $f \cdot \mu$, where $f$ is a locally $\mu$-integrable positive function, it is necessary and sufficient that every $\mu$-negligible compact set be $\nu$-negligible. If $f'$ is a second locally $\mu$-integrable function such that $\nu = f' \cdot \mu$, then $f = f'$ locally $\mu$-almost everywhere.

The condition is obviously necessary (Prop. 2). Conversely, suppose that every $\mu$-negligible compact set is $\nu$-negligible. Let us introduce a crushing $(K_\alpha)_{\alpha \in A}$ of T for the measure $\mu + \nu$ and let us set $N = T - \bigcup_{\alpha \in A} K_\alpha$. It is clear that $(K_\alpha)_{\alpha \in A}$ is a crushing for $\mu$ and for $\nu$, and Prop. 9 of §1, No. 8 therefore implies the following relations for every $g \in \mathscr{F}_+$:

Consider a compact set $C \subset K_\alpha$ that is $\mu_{K_\alpha}$-negligible; then $C$ is locally $\mu$-negligible, hence locally $\nu$-negligible, and finally $\nu_{K_\alpha}$-negligible by the definition of $\nu$. It then follows from the Lebesgue–Nikodym theorem (Ch. V, §5, No. 5, Th. 2) that $\nu_{K_\alpha}$ admits a density $f_\alpha$ with respect to $\mu_{K_\alpha}$. Let $f$ be the function that coincides with $f_\alpha$ on each of the sets $K_\alpha$, and with 0 on $N$; the function $f$ is $\mu$-measurable (Ch. IV, §5, No. 10, Prop. 16), and for every function $g \in \mathscr{F}_+$ one has, by the above relations and Prop. 3 of Ch. V, §5, No. 3,

It then follows first of all that $f$ is locally $\mu$-integrable: if $x$ is a point of T, and if $V$ is a neighborhood of $x$ such that $\nu^\bullet(V) < +\infty$, then $\mu^\bullet(f \varphi_V) < +\infty$. Next, Prop. 2 shows that the measures $\nu$ and $f \cdot \mu$ have the same essential upper integral. They are therefore equal (§1, No. 2, Cor. of Prop. 2). The uniqueness of $f$ being obvious based on the case of compact spaces, the proposition is established.

Remark 3). — The theory of integration with respect to a measure $\nu = f \cdot \mu$ reduces at once to the theory treated in Ch. V. For, let $(K_\alpha)_{\alpha \in A}$ be a crushing of T for $\mu$, hence for $\nu$, and let $T'$ be the locally compact space defined in the Scholium of §1, No. 8; we can associate to $\mu$ (resp. to $\nu$) a measure $\mu'$ (resp. $\nu'$) on $T'$, in such a way that the measurable functions, the essentially integrable functions with values in a Banach space, and the essential upper integrals of positive functions, are the same for $\mu$ and $\mu'$ (resp. for $\nu$ and $\nu'$). The function $f$ is therefore $\mu'$-measurable; it is locally $\mu'$-integrable, because $T'$ is locally compact, and a compact subset of $T'$ intersects only a finite number of the compact sets $K_\alpha$ ($\alpha \in A$). Finally, the relation ${\nu'}^\bullet(g) = \nu^\bullet(g) = \mu^\bullet(fg) = {\mu'}^\bullet(fg)$ proves that $\nu' = f \cdot \mu'$ (Ch. V, §5, No. 3, Prop. 3). We leave to the reader the task of transcribing the results of Ch. V, §5.