Littlewood’s three principles (2)
The second principle asserts that every function is nearly continuous. Now we investigate the meaning of “nearly” in terms of mathematics.
Theorem 2 (Lusin) Suppose f is measurable and finite valued on E with E of finite measure. Then for every there exists a closed set with
and such that is continuous.
Proof: Since by assumption, is measurable and finite valued on E, we can find a sequence of step functions, such that almost everywhere. Note that each is a step function which means it is continuous almost everywhere as it contains only at most countably many discontinuities which have Lebesgue measure 0. So without loss of generality, say for each we can find sets with so that is continuous outside . Now to proceed, we need something extra which is obtained from the third littlewood’s principle, i.e., Egrovff’s theorem. It enables us to find a set with such that uniformly on . Then for the integer N such that , we consider
For every , the step function is continuous on ; therefore, the uniform limit is also continous on . To complete the proof, notice that is also measurable so we can approximate from below by a closed set such that which gives us the desired .