One way to partially answer your last question might be the following. To each $f\in L^2(a,b)$, first associate its Lebesgue primitive $F(x)=\int_a ^x f(t)dt$, then define $Tf$ as one of the four Dini derivatives of $F$, e.g. $$ Tf(x)=\limsup _{h\to 0^+}h^{-1}(F(x+h)-F(x)).$$ Then $Tf=Tg$ everywhere if $f=g$ almost everywhere, $Tf=f$ almost everywhere, and $Tf$ is continuous if $f$ is equivalent to a continuous function. Thus the map $T$ associates to all members of a class of equivalence in $L^2$ the same function, which is the continuous representative of the class when it exists. An additional advantage is that the method is 'constructive'.
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One way to partially answer your last question might be the following. To each $f\in L^2(a,b)$, first associate its Lebesgue primitive $F(x)=\int_a ^x f(t)dt$, then define $Tf$ as one of the four Dini derivatives of $F$, e.g. $$ Tf(x)=\limsup _{h\to 0^+}h^{-1}(F(x+h)-F(x)).$$ Then $Tf=Tg$ everywhere if $f=g$ almost everywhere, $Tf=f$ almost everywhere, and $Tf$ is continuous if $f$ is equivalent to a continuous function. |
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