I'm reading the construction of Bochner integral from this Wikipedia page.
The construction seems circular and thus problematic to me. We are calculating an integral containing $f$, i.e. $$\lim_{n \rightarrow \infty} \int_{X}\left\| \color{blue}{f}-s_{n}\right\|_{B} \, d \mu=0$$ before we define the integral of $f$, i.e. $$\int_{X} f \, d \mu=\lim _{n \rightarrow \infty} \int_{X} s_{n} \, d \mu$$
Could you please verify my observation? Thank you so much!
Note that the function $x\mapsto \|f-s_n\|_B$ is a function in $X\to\mathbb{R}$ or $X\to\mathbb{C}$ as opposed to the function $f$, which has values in some Banach space.
They are extending the integral of real or complex valued functions, to some functions that return values in a Banach space.