Integral in lemma 1.20 in Takesaki's book Consider the following fragment from Takesaki's book "Theory of operator algebras II" chapter VI lemma 1.20:

How should the integral in (36) defined? I imagine it is some limit
$$\lim_{s\to \infty} \int_{0}^s + \lim_{t\to -\infty}\int_t^0$$
but then the question is how the integral
$$\int_a^b f(t)dt$$
is defined where $f:[a,b]\to A$ is a Banach-space valued function. Is this a Bochner integral? Or simply the usual Riemann integral? What ensures that such an integral exists?
 A: If the function $f:[a,b]\to A$ is continuous, then you can define the integral as a Riemann integral. If you look at how you prove that a Riemann integral  of a continuous function exists, you'll see that all you use regarding the codomain (that is $\mathbb R$ in the calculus case) is that it is a Banach space; that is, you take linear combinations and you use the absolute value for estimates, and completeness for the limit to exist. Everything works the same in a Banach space, you don't have to change the proofs at all. If you were to make the domain something else, say a Banach space, then things get more complicated and you need measure theory (which is where the Bochner integral fits in) because you need to somehow measure the size of your partition.
As for the improper integral, you can define it as you say. A Banach space is a place where Cauchy sequences are convergent, so if the sequence of integrals $\int_0^n f$ is Cauchy, then the limit will exist. And you always have the estimate
$$
\Big\|\int_a^bf(t)\,dt\Big\|\leq\int_a^b\|f(t)\|\,dt
$$
by the triangle inequality, which can be used to show that the sequence $\{\int_0^nf\}$ is Cauchy.

The Residue Theorem says that
$$\tag1
\oint_\gamma f(w)\,dw=2\pi i \sum_{z\in P_0} \operatorname{Res}(f,z)\,\operatorname{Ind}_\gamma(z).
$$
When $f$ is operator valued holomorphic/meromorphic, we consider a bounded linear functional $\varphi$ and then $\varphi\circ f$ is holomorphic/meromorphic. So
$$\tag2
\oint_\gamma (\varphi\circ f)(w)\,dw=2\pi i \sum_{z\in P_0} \operatorname{Res}(\varphi\circ f,z)\,\operatorname{Ind}_\gamma(z).
$$
Since $(\varphi\circ f)'=\varphi\circ f'$, the definition of residue works for $f$ and $\operatorname{Res}(\varphi\circ f,z)=\varphi(\operatorname{Res}(f,z))$. Hence
$$\tag3
\varphi\Bigg(\oint_\gamma f(w)\,dw\Bigg)=\varphi\Bigg(2\pi i \sum_{z\in P_0} \operatorname{Res}(f,z)\,\operatorname{Ind}_\gamma(z)\Bigg).
$$
And, as $\varphi$ was arbitrary, we get
$$\tag4
\oint_\gamma f(w)\,dw=2\pi i \sum_{z\in P_0} \operatorname{Res}(f,z)\,\operatorname{Ind}_\gamma(z).
$$
for operator-value meromorphic $f$.
