A real L^1 density defines a finite signed measure with its canonical Hahn and Jordan data

Put $\alpha (E)={\int }_E{f}^{+}d\mu$ and $\beta (E)={\int }_E{f}^{-}d\mu$. By [L1]–[L3] these are positive measures, and their total masses are finite because $f$ is integrable. By [L2], $\nu (E)=\alpha (E)-\beta (E)$ is finite. For disjoint measurable $E_n$, the series ${\sum }_n\alpha (E_n)$ and ${\sum }_n\beta (E_n)$ have finite sums, so they can be subtracted termwise to obtain countable additivity of $\nu$; moreover ${\sum }_n\mathrm{\mid }\nu (E_n)\mathrm{\mid }\le \alpha (X)+\beta (X)<\mathrm{\infty }$. Also $\nu (\mathrm{\varnothing })=0$. Thus [L6] makes $\nu$ a finite signed measure.

By [L3], $P=\{f>0\}$ and $N=\{f\le 0\}$ are measurable and partition $X$. The density $f^{-}$ vanishes on $P$ and $f^{+}$ vanishes on $N$, so on measurable subsets of $P$ one has $\nu =\alpha \ge 0$, and on subsets of $N$ one has $\nu =-\beta \le 0$. These signs prove directly that $(P,N)$ is a Hahn partition. Only the conditional comparison in [L4] is needed to say that every other supplied Hahn partition differs by a signed-null set.

Since $\alpha$ vanishes on $N$ and $\beta$ vanishes on $P$, the conditional construction in [L4] identifies them with the unique Jordan measures: $\nu (E\cap P)=\alpha (E)$ and $-\nu (E\cap N)=\beta (E)$. No arbitrary Hahn partition is selected.

For any countable measurable partition $E={⨆}_j{E}_j$, one has ${\sum }_j\mathrm{\mid }\nu (E_j)\mathrm{\mid }\le {\sum }_j(\alpha (E_j)+\beta (E_j))=\alpha (E)+\beta (E)$. Conversely, the explicit partition $E\cap P,E\cap N$ (with empty later parts) has variation sum exactly $\alpha (E)+\beta (E)$ by step 2.1. Taking the supremum in [L6] proves $\mathrm{\mid }\nu \mathrm{\mid }(E)=\alpha (E)+\beta (E)={\int }_E\mathrm{\mid }f\mathrm{\mid }d\mu$ by [L2] and [L5]. Together with steps 1.1–3.1 this proves every assertion without using the general Hahn/Jordan existence clauses.