Suppose that f has a measurable domain and is continuous except at a finite number...

Erin Lozano

Erin Lozano

Answered

2022-07-01

Suppose that f has a measurable domain and is continuous except at a finite number of points. Is f neccessarily measurable?
Let A be the set of points where f is not continuous. Now since A is finite we have that m ( A ) = 0. Now as continuous maps are measurable if E is the measurable domain, then f defined on E A is measurable.
Now I have a theorem that states
For a measurable subset D of E, f is measurable on E if and only if f D and f E D are both measurable.
f E A being measurable follows from the fact that f is continuous on E A, but how can I show that f is measurable on A? I only know that the measure of A is zero, but nothing about the behavior of f there?

Answer & Explanation

Sydnee Villegas

Sydnee Villegas

Expert

2022-07-02Added 22 answers

Actually f is Borel measurable.
Hints: We may as well take the domain to be the whole real line (by looking at f χ D ). Let f n ( x ) = f ( [ n x ] n ). Check that each f n is Borel measurable. Let g ( x ) = lim sup f n ( x ). Then g is Borel measurable and f=g except at finite number of points. Can you finish? [Inverse images of a Borel set under f and g differ only by a finite

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