Vector-valued integration with respect to a complex measure Rudin's Functional Analysis defines the vector-valued integration in Definition 3.26 as:

Suppose $\mu$ is a measure on a measure space $Q$, $X$ is a topological vector space on which $X^*$ separates points, and $f$ is a function from $Q$ into $X$ such that the scalar functions $\Lambda f$ are integrable with respect to $\mu$, for every $\Lambda\in X^*$; note that $\Lambda f$ is defined by
\begin{equation}
(\Lambda f)(q) = \Lambda(f(q)) \quad (q\in Q).
\end{equation}
If there exists a vector $y\in X$ such that
\begin{equation}
\Lambda y = \int_Q(\Lambda f)d\mu
\end{equation}
for every $\Lambda\in X^*$, then we define
\begin{equation}
\int_Qfd\mu = y.
\end{equation}

Then, an existence theorem of the vector-valued integration is provided in Theorem 3.27 as:

Suppose
(a) $X$ is a topological vector space on which $X^*$ separates points, and
(b) $\mu$ is a Borel probability measure on a compact Hausdorff space $Q$.
If $f:Q\to X$ is continuous and if $\overline{co}(f(Q))$ is compact in $X$, then the integral
\begin{equation}
y = \int_Qfd\mu
\end{equation}
exists, in the sense of Definition 3.26.
Moreover, $y\in\overline{co}(f(Q))$.

Remark to Theorem 3.27 says

If $\nu$ is any positive Borel measure on $Q$, then some scalar multiple of $\nu$ is a probability measure. The theorem therefore holds (except for its last sentence) with $\nu$ in place of $\mu$. It can then be extended to real-valued Borel measures (by the Jordan decomposition theorem) and (if the scalar field of $x$ is $C$) to complex ones.

When I read Definition 3.26, I understood $\int_Q(\Lambda f)d\mu$ as the integration of a complex function $\Lambda f$ with respect to a complex measure $\mu$, defined in the section 6.18 of Rudin's Real and Complex Analysis as:

If $\mu$ is a complex Borel measure, Theorem 6.12 asserts that there is a complex Borel function $h$ with $|h| = 1$ such that $d\mu = hd|\mu|$. It is therefore reasonable to define integration with respect to a complex measure $\mu$ by the formula
\begin{equation}
\int fd\mu = \int fhd|\mu|.
\end{equation}

However, Remark to Theorem 3.27 seems to imply
\begin{equation*}
\int_Q(\Lambda f)d\mu = \left(\int_Q(\Lambda f)d\Re(\mu)^+ - \int_Q(\Lambda f)d\Re(\mu)^-\right) + i\left(\int_Q(\Lambda f)d\Im(\mu)^+ - \int_Q(\Lambda f)d\Im(\mu)^-\right)
\end{equation*}
where $\Re(\mu) = \Re(\mu)^+ - \Re(\mu)^-$ and $\Im(\mu) = \Im(\mu)^+ - \Im(\mu)^-$ are the Jordan decompositions of the real and imaginary parts of $\mu$.
Since the polar decomposition $d\mu = hd|\mu|$ does not guarantee continuity of $h$, I guess that Theorem  3.27 cannot be used to prove the existence of $y\in X$ satisfying
\begin{equation}
\Lambda y = \int_Q(\Lambda f)hd|\mu| = \int_Q(\Lambda(hf))d|\mu|
\end{equation}
for all $\Lambda\in X^*$ where $(\Lambda(hf))(q) = \Lambda(h(q)f(q))$.
So, I am a bit confused with two different definitions of the Lebesgue integration with respect to a complex measure.
Are they same? If not, what are the major differences between them and is there any reference that compares them?
 A: By following the comment of copper.hat, I could prove that they are same.

If $\mu$ is a complex measure on a $\sigma$-algebra $\mathfrak{M}$ in $X$, $d\mu = hd|\mu|$ is the polar decomposition of $\mu$, $\mu = \Re(\mu) + i\Im(\mu) = \mu_1 - \mu_2 + i(\mu_3 - \mu_4)$ is the Jordan decompositions of the real and imaginary parts of $\mu$, $f\in L^1(|\mu|)$, and $E\in\mathfrak{M}$, then
\begin{equation}
\int_Efhd|\mu| = \left(\int_Efd\mu_1 - \int_Efd\mu_2\right) + i\left(\int_Efd\mu_3 - \int_Efd\mu_4\right).
\tag{1}
\end{equation}

proof: Since $|h| = 1$ and $\mu_j(A) \le |\mu|(A)$ for every $j$ and $A\in\mathfrak{M}$, $f\in L^1(|\mu|)$ implies $fh\in L^1(|\mu|)$ and $f\in \bigcap_{j=1}^4L^1(\mu_j)$.
Thus, the left and the right sides of (1) are well defined.
Since $|\mu|$ is bounded and
\begin{align*}
    \int_E\chi_Ahd|\mu| &= \mu(E\cap A) \\
    &= \mu_1(E\cap A) - \mu_2(E\cap A) + i(\mu_3(E\cap A) - \mu_4(E\cap A)) \\
    &= \left(\int_E\chi_A\mu_1 - \int_E\chi_A\mu_2\right) + i\left(\int_E\chi_A\mu_3 - \int_E\chi_A\mu_4\right)
\end{align*}
for every characteristic function $\chi_A$ for $A\in\mathfrak{M}$, (1) holds for every simple measurable function.
Since simple measurable functions are dense in $L^1(|\mu|)$, there is a sequence $\{f_k\}$ in $L^1(|\mu|)$ such that (1) holds for each $f_k$ and $f_k \to f$ as $k\to\infty$ in $L^1(|\mu|)$.
Since $L^1(|\mu|) \subset \bigcap_{j=1}^4L^1(\mu_j)$, $f_k$ also converges to $f$ in $\bigcap_{j=1}^4L^1(\mu_j)$.
Therefore,
\begin{align*}
    &\left|\int_Efhd|\mu| - \left(\int_Efd\mu_1 - \int_Efd\mu_2\right) - i\left(\int_Efd\mu_3 - \int_Efd\mu_4\right)\right| \\
    &\le \int_E|f - f_k|d|\mu| + \sum_{j=1}^4\int_E|f-f_k|d\mu_j \\
    &\to 0 \quad (k\to\infty).
\end{align*}
