9.9. SOME IMPORTANT GENERAL THEORY 213

By Theorem 3.12.5, there exists hi ∈Cc(X) such that

hi ≺Vi,n

∑i=1

hi(x) = 1 on spt( f ).

Now note that for each i, f (x)hi(x)≤ hi(x)(ti + ε). Therefore,

L f = L(n

∑i=1

f hi)≤ L(n

∑i=1

hi(ti + ε)) =n

∑i=1

(ti + ε)L(hi)

=n

∑i=1

(|t0|+ ti + ε)L(hi)−|t0|L

(n

∑i=1

hi

).

Now note that |t0|+ ti + ε ≥ 0 and so from the definition of µ and Claim 2 of the proof ofTheorem 8.8.2, this is no larger than

n

∑i=1

(|t0|+ ti + ε)µ(Vi)−|t0|µ(spt( f ))≤n

∑i=1

(|t0|+ ti + ε)(µ(Ei)+ ε/n)−|t0|µ(spt( f ))

≤ |t0|

µ(spt( f ))︷ ︸︸ ︷n

∑i=1

µ(Ei)+ε

nn |t0|+∑

itiµ (Ei)+∑

iti

ε

n+∑

iεµ (Ei)+

ε2

n−|t0|µ(spt( f ))

≤ ε |t0|+ ε (|t0|+ |b|)+ εµ(spt( f ))+ ε2 +∑

itiµ (Ei)

≤ ε |t0|+ ε (|t0|+ |b|)+2εµ(spt( f ))+ ε2 +

n

∑i=1

ti−1µ(Ei)

≤ ε (2 |t0|+ |b|+2µ(spt( f ))+ ε)+∫

f dµ

Since ε > 0 is arbitrary, L f ≤∫

f dµ for all f ∈Cc(X), f real. Hence equality holds becauseL(− f ) ≤ −

∫f dµ so L( f ) ≥

∫f dµ . Thus L f =

∫f dµ for all f ∈ Cc(X). Just apply the

result for real functions to the real and imaginary parts of f . ■

9.9 Some Important General Theory9.9.1 Eggoroff’s TheoremYou might show that a sequence of measurable real or complex valued functions convergeson a measurable set. This is Proposition 8.1.7 above. Eggoroff’s theorem says that if theset of points where a sequence of measurable functions converges is all but a set of measurezero, then the sequence almost converges uniformly in a certain sense.

Theorem 9.9.1 (Egoroff) Let (Ω,F ,µ) be a finite measure space, µ (Ω) < ∞ andlet fn, f be complex valued functions such that Re fn, Im fn are all measurable and alsothat limn→∞ fn(ω) = f (ω) for all ω /∈ E where µ(E) = 0. Then for every ε > 0, there existsa set, F ⊇ E, µ(F)< ε, such that fn converges uniformly to f on FC.