When we talk about $L^p$ space as a normed vector space, we use the norm
$$\| f \|_p = \left(\int_{E}{|f|^p d\mu}\right)^{1/p}.$$
But, the problem is that $\|f\|_p$ can be zero even if $f \not\equiv 0$ (i.e. if it is non-zero on a space of zero measure). So does this mean that when we talk about $L^p$ space we are really talking about the set of functions with finite $p$-norm modulo two functions being equivalent if they are agree everywhere except a set of measure zero?