Can we find two linearly independent index functions?

Question

Let $(X, \mathfrak{M}, \mu)$ be a measure space. Suppose that for some $p \in [1, \infty]$, we know that $\dim(L^p) \geq 2$. Can we guarantee the existence of 2 linearly independent index function $\chi_A, \chi_B \in L^p$ ? I.e. functions $\chi_A, \chi_B$ such that $$ a\chi_A + b\chi_B = 0\text{ a.e.}\implies a=b=0 $$

Answer 1

Yes. If there are no disjoint sets of positive measure the every measurable function would be almost surely constant and $L^{p}$ would be one dimensional. Otherwise $\chi_A$ and $\chi_B$ would be independent when A and B are disjoint sets of positive measure.