Page 377 - DMTH401_REAL ANALYSIS

Unit 31: The Integral of a Non-negative Function

where the first equality follows from h = 0 on E/A and the last line h £ f on E for all n. Taking Notes
n n
supremum over all such h’s, we get the desired inequality.
Theorem 2: Monotone Convergence Theorem

If {f } is an increasing sequence of non-negative measurable functions defined on E (increasing in
n
the sense that f £ f for all n on E) and f  f a.e. on E, then
n n+1 n
E ò f ­ E ò f
n
by which it means {j f } is an increasing sequence with limit ò f .
E n E
In symbol,

0 £ fn ­ f a.e. on E  f ­ E ò f
E ò
n
Proof:
f
E ò f £ liminf E ò f £ limsup f £ E ò ,
E ò
n
n
n n
the first inequality follows from Fatou’s Lemma, the last inequality follows from f £ f on E for
n
all n. Hence f ­ E ò f . (That f increases as n increases is immediate from monotonicity of such
E ò
E ò
n
n
integrals.)
Corollary: Extension of Fatou’s lemma
If {f } is a sequence of non-negative measurable functions on E, then liminf f £ liminf f .
n E ò n n n E ò n
Proposition: Suppose f is a non-negative measurable function defined on E such that ò f <  .
E
Then for all  > 0, there is a  > 0 such that
E ò f < 
whenever A  E with m(A) < .
Proof: The result clearly holds if f is bounded on E. Suppose now f is not necessarily bounded, we
­
see that (f Ù n) f so by Monotone Convergence Theorem
A ò f = lim A ò (f Ù n)
n
for all A  E. Note that by assumption ò f <  so both sides of the equality above are finite.
E
Hence if  > 0 is given, then there is a N such that ò f - (f Ù N) < 
A A ò .
Take  = /2N, we see that
+
+
A ò f £ A ò f - A ò f(f Ù N) A ò + (f Ù N) £  /2 Nm(A)£  /2 N < 
whenever A  E with m(A) < . So we are done.

31.2 Extended Real-valued Integrable Functions

Here we integrated non-negative measurable functions, and we wish to drop the non-negative
requirement. Recall that it is a natural requirement that our integral be linear, and now we can
integrate a general non-negative measurable function, so it is tempting to define the integral of
+ -
a general (not necessarily non-negative) measurable function f to be f ò - f ò where f+ = f V0
+
–
–
and f = (–f) V0, since f , f are non-negative measurable and they sum up to f. But it turns out that
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