Does almost everywhere equality means equality on the quotient space?

Question

I always have the confusion that L1 norm is only a seminorm on the L1 space (because L1 norm of a function = 0 iff the function is almost everywhere 0, but not exactly 0), but always called a 'norm'. So is the name 'L1 space' with a.e. equality actually a quotient space of L1 mod all the function with integral 0? And the same with Lp, excluding infinite p.

Answer 1

You're are right except for infinite $p$ this will also work. L-p norm is usually defined for space of L-p Lebesgue integrable functions. If you want such norm to be defined without using Lebesgue measure, you could choose space of $C^1[0,1]$. In this case, the function in your example will have positive norm. When Lebesgue measure is used, set of measure zero doesn't matter any more. And you are right about norm are equivalent on classes as you described. For $L^\infty$ with Lebesgue integration, the essential supremum norm is defined as $essup\{b\in\mathbb{R}|m(f^{-1}(b,\infty))=0\}$. This ignores the null-set difference you mentioned.