Question: I came accross this question in a practice test for finance. 
[![enter image description here](https://i.stack.imgur.com/LK9al.png)](https://i.stack.imgur.com/LK9al.png)

They asked us to find the value of the company's cash flow from operations in 2012 (blue line) which decreased by 2.3%. We have only the value of cash flow from operations in 2011 which according to this shape is 10 million. The answer guide indicates that it is 30 million and not 10. How is it 30 million? the blue dot on 2011 indicates it is 10 million.

Answer: 
Write $[f]$ the equivalent class of $f$ and define $\|[f]\|\_{L^{p}}=\left(\displaystyle\int|f|^{p}d\mu\right)^{1/p}$, one can show that any representative of $f$, say, $f=g$ a.e. then $\left(\displaystyle\int|f|^{p}d\mu\right)^{1/p}=\left(\displaystyle\int|g|^{p}d\mu\right)^{1/p}$, so the definition of $\|[f]\|\_{L^{p}}$ is okay. Now if $\|[f]\|\_{L^{p}}=0$, then $\displaystyle\int|f|^{p}d\mu=0$, as $|f|^{p}\geq 0$, it is a standard fact that $|f|=0$ a.e. then $f=0$ a.e. so $[f]=[0]$.
The point is that $||f||\_p$ is not a norm on functions because, as you say, $||f||\_p = 0$ would imply $f$ identically $0$. However, if you work for example on $\mathbb{R}$, the function
$$f \colon \mathbb{R} \rightarrow \mathbb{R}, f(0) = 1, f(x) = 0 \text{ when } x \neq 0$$
satisfies $||f||\_p = 0$, and it is not identically $0$! So one identifies functions which coincide almost everywhere, so that $f$ in my example is actually equivalent to the function identically $0$, which has norm $0$. This way you define $|| \cdot ||\_p$ on equivalence classes of functions which is a well-defined norm.