$L^2(\mathbb R, \mu) $ a finite dimensional space.

Question

Hi I find the following exercise. Honestly I'm not sure about my "answer", is incredible simple.t I don't know if make sense (in what part is necessary to use $L^2$?). I'd appreciate if someone can clarify the ideas of the exercise.Thank you

Prove that if a measure $\mu$, defined on $\mathscr B (\mathbb R)$ it has a finite number of increasing points, then $ L^2 $ is a finite dimensional space.

I'm not sure of this, but my idea is as follows. Let $K=\{c_1, \ldots, c_n\}$ the support of $\mu$. Let $f$ in $ L^2 (\mathbb R)$, so $f(x) = \sum_{1\le k \le n} f(c_k) \chi_{c_k} (x)$ holds a.e.-$\mu$. Thus for each $f$ in $L^2$ it can be written as a linear combination of $\{\chi_{c_k}: 1\le k \le n\}$ so is a base to $L^2$ an so is finite dimensional space.

Answer 1

As pointed out in the comments, the attempt is good. Furthermore, the argument works for $\mathbb L^p$ where $1\leqslant p<+\infty$.