While trying to prove the uniqueness property ($⟨x,x⟩=0$ iff $x=0$), I understand that I would arrive at a point like this: $$⟨f,f⟩= \int |f(x)|^2 dx=0.$$ Following which I have to somehow prove that $f$ is identically zero.
However, from what I know, (and currently my understanding of lebesgue intergration is loose and informal, sorry) since Lebesgue integrable functions can be discontinuous, it's not possible to conclude that $\int|f(x)|^2 dx$ implies $f(x) = 0$ for all $x \in [a, b].$
Is there something I'm missing/getting wrong? How would we go about proving the uniqueness criteria?
Thank you!
In general, if $(X,\mathcal{A}, \mu)$ is a measured space and $f : X \rightarrow \mathbb{R}$ a $\mathcal{A}$-measurable application satisfying $$\int |f|d\mu = 0,$$ then $f=0$ almost everywhere. Since in $L^2$, two functions which are equals almost everywhere are equals, you have $f=0$.