Multivariable Advanced Calculus

196 THE ABSTRACT LEBESGUE INTEGRALFrom the first part of the lemma, there exists a sequence {f k } such thatK k ≺ f k ≺ V k .Then f k (x) converges to X E (x) a.e. because if convergence fails to take place, then xmust be in infinitely many of the sets V k \ K k . Thus x is in∩ ∞ m=1 ∪ ∞ k=m V k \ K kand for each pµ (∩ ∞ m=1 ∪ ∞ k=m V k \ K k ) ≤ µ ( ∪ ∞ k=pV k \ K k)≤ 0 be given. Then there existsg ∈ C c (Y ) such that∫Y|f (x) − g (x)| dµ < ε.Proof: By considering separately the positive and negative parts of the real andimaginary parts of f it suffices to consider only the case where f ≥ 0. Then by Theorem7.7.12 and the monotone convergence theorem, there exists a simple function,s (x) ≡p∑c m X Em (x) , s (x) ≤ f (x)m=1such that∫|f (x) − s (x)| dµ < ε/2.By Lemma 8.10.2, there exists {h mk } ∞ k=1 be functions in C c (Y ) such that∫lim |X Em − f mk | dµ = 0.k→∞ YLetp∑g k (x) ≡ c m h mk .m=1