I ##L^2## - space equivalence classes and norm

cianfa72
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TL;DR Summary
##L^2##-space as set of equivalence classes of squared integrable measurable functions
##L^2##-space is defined as equivalence classes on the set ##\mathcal L^2## of squared integrable measurable functions ##f## defined on the measure space ##(\Omega, \mathcal A, \mu)##.

The equivalence relation ##\sim## is: ##f \sim g## iff ##f=g## almost everywhere (a.e.).

Prove that the above is equivalent to ##||f|| = ||g||##. The norm ##|| \, . ||## is defined as $$\int_{\Omega} {| \, . |}^2 \, d{\mu}$$
My attempt
Consider the map ##|f| - |g|## that is measurable since ##f,g## are. ##|f| - |g| \neq 0## on a subset of the (measurable) set ##A## where ##(f - g)## is, therefore such subset (that is measurable itself) has measure 0. Hence $$\int_{\Omega} {|f| - |g|} \, d{\mu} = 0$$ i.e. by linearity $$\int_{\Omega} {|f|} \, d{\mu} = \int_{\Omega} {|g|} \, d{\mu}$$
Same argument applies to ##|f|^2 - |g|^2##.

Does the above make sense ? Thanks.
 
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cianfa72 said:
TL;DR Summary: ##L^2##-space as set of equivalence classes of squared integrable measurable functions

##L^2##-space is defined as equivalence classes on the set ##\mathcal L^2## of squared integrable measurable functions ##f## defined on the measure space ##(\Omega, \mathcal A, \mu)##.

The equivalence relation ##\sim## is: ##f \sim g## iff ##f=g## almost everywhere (a.e.).

Prove that the above is equivalent to ##||f|| = ||g||##. The norm ##|| \, . ||## is defined as $$\int_{\Omega} {| \, . |}^2 \, d{\mu}$$

This is not a norm: you need to take the square root of it for it to satisfy the property \|af\| = |a|\|f\| for constant a.

I think the equivalence that you are looking for is that f = g a.e. is equivalent to \|f - g\| = 0.

f = g almost everywhere implies \|f\| = \|g\|, but the converse does not hold: Take f = 1, g = -1 and \Omega = [0,1] for a counterexample.

One of the axioms of a norm is that \|f\| = 0 if and only if f is the zero vector; for functions under pointwise addition the zero vector is the function which is identically zero. But for the L^2 norm, a function which is different from zero on a set of measure zero - and is therefore distinct from the zero vector - will have zero norm. We fix this by working instead with equivalance classes, with functions defined to be equivalent if and only if they differ from each other by a function with zero norm, i.e. f \sim g \quad\Leftrightarrow \quad \|f - g\| = 0.
 
pasmith said:
This is not a norm: you need to take the square root of it for it to satisfy the property \|af\| = |a|\|f\| for constant a.
Ok yes, definitely. It is actually $$\left( \int_{\Omega} {| \, . |}^2 \, d{\mu} \right)^{\frac 1 2}$$
pasmith said:
f = g almost everywhere implies \|f\| = \|g\|, but the converse does not hold: Take f = 1, g = -1 and \Omega = [0,1] for a counterexample.
Ok, yes.

pasmith said:
One of the axioms of a norm is that \|f\| = 0 if and only if f is the zero vector; for functions under pointwise addition the zero vector is the function which is identically zero. But for the L^2 norm, a function which is different from zero on a set of measure zero - and is therefore distinct from the zero vector - will have zero norm. We fix this by working instead with equivalance classes, with functions defined to be equivalent if and only if they differ from each other by a function with zero norm, i.e. f \sim g \quad\Leftrightarrow \quad \|f - g\| = 0.
Ok, in the specific case the latter equivalence boils down to ##{|f - g|}^2 \neq 0## if and only if ##f \neq g##. Therefore both hold true on the same (measurable) set ##A## with measure 0. This is true for any p-norm.
 
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pasmith said:
We fix this by working instead with equivalance classes, with functions defined to be equivalent if and only if they differ from each other by a function with zero norm, i.e. f \sim g \quad\Leftrightarrow \quad \|f - g\| = 0.
BTW, I'd say they are defined to be equivalent iff they differ each other by a function with zero "semi-norm" i.e. w.r.t. the map that will be promoted to a full norm when considering the set of equivalence classes being defined.
 
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cianfa72 said:
BTW, I'd say they are defined to be equivalent iff they differ each other by a function with zero "semi-norm" i.e. w.r.t. the map that will be promoted to a full norm when considering the set of equivalence classes being defined.

Technically yes, although in general usage one refers to \|\cdot\|_p : \mathcal{L}^p \to [0, \infty) : f \mapsto \left(\int_\Omega |f|^p\,d\mu\right)^{1/p} as a norm and writes \|f\|_p for what is technically \|[f]\|_p. It's also possible to find that both the set of functions for which \int_{\Omega} |f|^p\,d\mu exists and the set of equivalence classes of such functions are denoted by \mathcal{L}^p(\Omega).
 

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