Definition 6

One calls complex measure on $T$ every continuous linear form on the complex vector space $\mathscr{K}_C(T)$.\footnote{Cf. Ch. III, §1, No. 3, Def. 2.}

The space $\mathscr{M}_C(T)$ of complex measures on $T$ is thus the dual of the Hausdorff locally convex space $\mathscr{K}_C(T)$.

If $m$ is a complex measure on $T$, its restriction to $\mathscr{K}(T)$ is a vectorial measure on $T$ with values in $\mathbf{C}$ (regarded as a vector space over $\mathbf{R}$);

m is determined by this restriction, since if $f = f_1 + i f_2 \in \mathscr{H}_\mathbf{C}(T)$, the real part $f_1$ and the imaginary part $f_2$ of $f$ are in $\mathscr{H}(T)$, and $m(f) = m(f_1) + i m(f_2)$. Conversely, for every vectorial measure $m_0$ on T with values in $\mathbf{C}$, the formula $m(f) = m_0(f_1) + i m_0(f_2)$ defines a complex measure $m$, the only one on T whose restriction to $\mathscr{H}(T)$ is $m_0$. We shall therefore henceforth identify a complex measure with its restriction to $\mathscr{H}(T)$; such a measure $m$ is of the form $m = \mu_1 + i \mu_2$, where $\mu_1$ and $\mu_2$ are two real measures on T, which are called, respectively, the real part and the imaginary part of $m$. The support of $m$ is the union of the supports of $\mu_1$ and $\mu_2$. One knows that $m$ is majorizable (No. 3, Cor. of Prop. 4); we shall call absolute value of $m$ the positive measure $|m|$ corresponding to the absolute value $|x_1 + i x_2| = \sqrt{x_1^2 + x_2^2}$ on $\mathbf{C}$. One has $|m| = (\mu_1^2 + \mu_2^2)^{1/2}$ (No. 4, Remark following Prop. 9),$^3$ and $|\mu_1| \leq |m|$, $|\mu_2| \leq |m|$, $|m| \leq |\mu_1| + |\mu_2|$; moreover, $m$ is a measure with base $|m|$, and one can write $m = h \cdot |m|$, where $h \in \mathscr{L}_\mathbf{C}^\infty(|m|)$ and $|h(t)| = 1$ locally almost everywhere for $|m|$ (No. 4, Prop. 9).$^4$ The supports of $m$ and $|m|$ are the same.

For every mapping $f$ of T into a complex Banach space F, essentially integrable with respect to $|m|$, one can define (No. 7) the integral of $f$ with respect to $m$ (corresponding to the R-bilinear mapping $(x, \lambda) \mapsto \lambda x$ of $F \times \mathbf{C}$ into F), which will be denoted $\int f \, dm$; it follows at once from the uniqueness property of Prop. 11 that (with the preceding notations) $\int f \, dm = \int f \, d\mu_1 + i \int f \, d\mu_2 = \int f h \, d|m|$. We therefore have $m(f) = \int f \, dm$ for $f \in \mathscr{H}_\mathbf{C}(T)$. We say that $f$ is essentially integrable with respect to $m$ if it is so for $|m|$;$^5$ mappings that are $m$-integrable, $m$-measurable, locally $m$-integrable, $m$-negligible or locally $m$-negligible are defined similarly. We write

instead of $\mathscr{L}_F^p(T, |m|)$ (resp. $\overline{\mathscr{L}}_F^p(T, |m|), \ L_F^p(T, |m|)$); these are complex vector spaces.

For $f$ to be $m$-integrable (resp. essentially $m$-integrable), it is necessary and sufficient that $f$ be integrable (resp. essentially integrable) with respect to each of the four measures $\mu_1^+, \mu_1^-, \mu_2^+, \mu_2^-$, by virtue of the inequalities between $|m|, |\mu_1|, |\mu_2|$ and the relations $|\mu_k| = \mu_k^+ + \mu_k^-$ (Ch. V, §2, No. 2, Prop. 3 and its Cor. 1).

If $f$ is essentially $m$-integrable (resp. $m$-integrable), then $|f|$ is essen-

--- $^3$In particular, this definition of $|m|$ coincides with that in Ch. III, §1, No. 6 (cf. Ch. V, §5, No. 9). $^4$Cf. Ch. V, §5, No. 5, Cor. 3 of Th. 2 for a sharper statement. $^5$Cf. Ch. V, §1, remarks at the end of No. 3.

tially $|m|$-integrable (resp. $|m|$-integrable), and it follows from Prop. 11 that

Let F and G be two complex Banach spaces, $u$ a continuous linear mapping of F into G. If $\mathbf{f}$ is an essentially $m$-integrable (resp. $m$-integrable) mapping of T into F, then $u \circ \mathbf{f}$ is essentially $m$-integrable (resp. $m$-integrable) and $\int (u \circ \mathbf{f})\, dm = u(\int \mathbf{f}\, dm)$; this follows at once from the foregoing and the analogous proposition for essentially $|m|$-integrable functions (Ch. IV, §4, No. 2, Th. 1 and Ch. V, §1, No. 3, Lemma and Def. 3).

Let $m$ be a complex measure on T and let $h$ be a locally $m$-integrable complex function. For every function $f \in \mathscr{K}_C(T)$, the function $fh$ is $m$-integrable and the mapping $f \mapsto \int fh\, dm$ is a complex measure, denoted $h \cdot m$ and called the measure with density $h$ with respect to $m$. If $m = g \cdot |m|$, it is clear that $h \cdot m = hg \cdot |m|$; since, moreover, $|g(t)| = 1$ locally almost everywhere for $|m|$, for $\mathbf{f}$ to be essentially integrable for $n = h \cdot m$ it is necessary and sufficient that $fh$ be essentially $m$-integrable, in which case $\int \mathbf{f}\, dn = \int (\mathbf{f}h)\, dm$. Moreover, $|h \cdot m| = |h| \cdot |m|$. We again say that every measure of the form $h \cdot m$ is a complex measure with base $m$; two complex measures $m, m'$ are said to be equivalent if each has a density with respect to the other, or, what amounts to the same, if $m' = h \cdot m$ with $h$ locally $m$-integrable and $h(t) \neq 0$ locally almost everywhere for $|m|$. It is clear that $m$ and $|m|$ are equivalent and that, for $m$ and $m'$ to be equivalent, it is necessary and sufficient that $|m|$ and $|m'|$ be so.

If $m$ and $m'$ are two complex measures on T, and $\mathbf{f}$ is a function with values in a complex Banach space F, essentially integrable (resp. integrable) for both $m$ and $m'$, then, for any complex numbers $\lambda$ and $\lambda'$, $\mathbf{f}$ is essentially integrable (resp. integrable) for $\lambda m + \lambda' m'$, and

For, this follows from the Cor. of Prop. 11 of No. 7.

In addition, it follows from the definitions that

One calls conjugate measure of a complex measure $m$ the complex measure $\overline{m}$ defined by $\overline{m}(f) = m(\overline{f})$ for $f \in \mathscr{K}_C(T)$. If $m = \mu_1 + i \mu_2$, where $\mu_1$ and $\mu_2$ are real measures, then $\overline{m} = \mu_1 - i \mu_2$ and $|\overline{m}| = |m|$; if $m = h \cdot |m|$ then $\overline{m} = \overline{h} \cdot |m|$. If $f$ is an essentially $m$-integrable (resp.

m-integrable) function with complex values, then $\overline{f}$ is essentially $\overline{m}$-integrable (resp. $\overline{m}$-integrable) and