Vector-valued function space Traditionally, Functional Analysis studies the space of measurable functions of the form $f:X \rightarrow \mathbb{R}$ where $X$ is a mesurable space, e.g. $L^2([0,1])$. I am interested to study the properties of a space of functions of the form $\{f:[0,1] \rightarrow [0,1]^r \, | \, f \mbox{ is measurable } \}$ and $r$ is positive integer.
However I haven't found any reference about the space if valued-vector functions.
I wonder:
Does it exist reference to study this topic? 
Why there are few references about the topic.  
Thanks in advance. 
 A: A separable Hilbert space with orthonormal basis $\{ e_n \}$ is much easier to deal with than the general case. And your problem falls into that category.
For example, if
$$
                      f : [0,1]\rightarrow\mathscr{H}
$$
is weakly measurable, meaning that $\langle f(x),h\rangle$ is measurable for all $h\in\mathcal{H}$, then $\|f(t)\|^2 = \sum_{n}|\langle f(t),e_n\rangle|^2$ is measurable as well. Then the requirement that $\int_{0}^{1}\|f(t)\|^2dt < \infty$ gives a Hilbert norm:
$$\|f\|^2=\int_{0}^{1}\|f(t)\|^2dt.$$
You can define an integral $\int_{0}^{1}f(t)dt$ as the unique vector $\int_{0}^{1}f(t)dt$ such that
$$
    \int_{0}^{1}\langle f(t),x\rangle dt = \left\langle \int_{0}^{1}f(t)dt,x\right\rangle, \;\;\; \tag{*}
$$
which makes sense by the Riesz representation theorem because
$$
    \left|\int_{0}^{1}\langle f(t),x\rangle dt\right|
   \le \int_{0}^{1}|\langle f(t),x\rangle|dt\\
  \le \int_{0}^{1}\|f(t)\|\|x\|dt \\
  \le \left(\int_{0}^{1}\|f(t)\|^2dt\right)^{1/2}\|x\| \\
   = \|f\|\|x\|.
$$
Using this weak definition of integral, you can derive the relevant properties of the integral by reducing to the scalar case. And you automatically have property (*) above.
