Integration in normed vector spaces

Question

Given an interval $I\subset \mathbb{R}$ and a normed vector space $X$, I want to know if I am able to define the Lebesgue space $L^p(I;X)$ of all $p$-integrable functions $f:I\to X$. I know that this can be done if $X$ is a Banach space (which leads to the notion of Bochner spaces), but for general normed vector spaces I don't know. So I would be grateful for some help and references if it is possible. Thanks.

Answer 1

No you can’t. Completeness is critical.

The problem is that for a sequence of simple functions $(s_n)$ such that $\lim\limits_{n \to \infty} \int \Vert f -s_n\Vert =0$ the sequence $\int s_n$ may not converge in $X$.