Approximate measurable sets by open and compact sets

Try

A measurable set can have a complicated boundary, yet its measure can be approximated from outside by open sets and from inside by compact sets. For a finite-measure set E and any ε>0, there are compact K and open O with K⊆E⊆O and m(O outside K)<ε. This is regularity. It turns abstract measurability into controlled geometric approximation without claiming E itself is open or closed.

Predict

Predict why finite measure matters when expressing the error as a difference. If m(E)=∞, subtracting m(E) from m(O) is meaningless when both are infinite. Local approximation still works on bounded pieces, and outer approximation can be formulated with a small excess set, but the elementary inner approximation to the total measure is stated here for finite m(E). The proof will explicitly truncate before subtracting.

Experiment

Compare a familiar finite set with its approximants.

Object	Choice for E=(0,1)	Measure
Compact inner set	[δ,1−δ],0<δ<1/2	1−2δ
Target	(0,1)	1
Open outer set	(−δ,1+δ)	1+2δ

The total gap is 4δ. For an arbitrary measurable E no equally simple interval description is promised, but the theorem supplies the same type of measure control.

Understand

First establish that the interval-cover outer measure makes Borel sets measurable. For a half-line ending at a, split every covering interval at a; arbitrarily small enlargements handle open endpoints, with a summable total enlargement. The two resulting cover costs add to the original cost plus an arbitrary error. Taking infima proves the Carathéodory splitting inequality for the half-line. Half-lines generate the Borel sigma-algebra, so all Borel sets are measurable. The resulting complete measure agrees with interval length and is called Lebesgue measure. Translation invariance follows by translating covers in both directions and the measurable-set criterion.

Apply

For finite-measure measurable E, choose a countable open-interval cover with total length below m(E)+ε. Its union O is open, contains E, and has measure at most the total. Since E is measurable and finite, m(O outside E)=m(O)−m(E)<ε. If E is bounded, put it inside a closed bounded interval I. Apply this outer approximation to I outside E and let K=I outside O. The set K is closed and bounded, hence compact; it lies in E and loses less than ε of its measure. For unbounded E of finite measure, first choose N with m(E outside[−N, N]) small by continuity from below, then apply the bounded construction with the remaining error budget.

Continue

An open subset of the real line is a countable disjoint union of open intervals: its connected components are intervals, and each contains a distinct rational point. This gives a useful finite-interval approximation after truncating a finite-measure open set. Consequently measurable indicators can later be approximated in integral norm by step functions and continuous functions. Regularity concerns closeness in measure, not equality of sets or uniform closeness of indicator functions. A tiny exceptional set can still contain important individual points, so applications must distinguish integral error from pointwise guarantees.

Continue to Separate null-set completion from arbitrary measurability.