cianfa72
- 2,784
- 293
- 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.
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.
Last edited: