How continuous are Baire 1 functions?

Given a locally bounded measurable function Rn -> R, define the essential modulus of continuity, M(f): Rn x R+ -> R by

M(f) (x, e) = sup {d >= 0| f[B_d (x) setminus A] in [f(x) – e, f(x) + e] for some null set A}.

It can be shown (details here) that M is well defined a.e. in the product measure and does not depend on choice of representative.

Is it true that any Baire 1 function R -> R (pointwise limit of continuous functions) has essential moc equal a.e. to that of a continuous function?

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Nevin Manimala

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