Page 379 - DMTH401_REAL ANALYSIS

Unit 31: The Integral of a Non-negative Function

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(6) A ò (f g) + A ò f + A ò g = A ò (f g) + A ò f + A ò g , Notes
which will be true if we can show
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(7) (f + g) + f + g = (f + g) + f + g +
a.e. on A because we can then use linearity of Section 3 to conclude that (6) is true. But (7) is
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clearly true a.e., because (f + g) – (f + g) = f + g = f – f + g – g a.e., all terms being finite a.e. This
completes our proof.
Finally we prove the important Generalized Lebesgue Dominated Convergence Theorem.
Theorem 3: If {f }, {g } are sequences of measurable functions defined on E, |f | £ g , f = lim f , g
n n n n n n
= liminf g and lim ò E g = E ò g <  , then lim ò E f exists and is equal to ò f .
n
n
n
n n n E
Proof: Since |f | £ g implies g ± f are non-negative measurable, we see that
n n n n
£
+
E ò
E ò g + E ò f = E ò liminf(g + f ) liminf (g + f ) = E ò g liminf f n
E ò
n
n
n
n
n n   n  
and similarly
£
+
E ò
E ò
E ò g - E ò f = E ò liminf(g - f ) liminf (g - f ) = E ò g liminf f n
n
n
n
n
n n n
So f £ liminf E ò f £ limsup f (note here we used the assumption that g <  ) and the desired
E ò
E ò
E ò
n
n
n
conclusion follows.
Corollary: Lebsegue Dominated Convergence Theorem
Suppose a sequence of measurable functions {f } defined on E converges pointwisely a.e. on E to
n
f. If|f | £ g on E for some integrable function g, then ò f n converges to f .
E ò
n E
A final word of remark: The idea of this section extends readily to complex-valued functions,
and the readers who are familar with general measure theory should find that the results in the
whole unit is valid on a general measure space without needing the slightest modification.
Self Assessment
Fill in the blanks:
1. For a non-negative measurable function f : E  [0, ] (where E is a set which may be of
finite or infinite measure), we define .................................... .
2. For non-negative ................................ vanishing outside a set of finite measure, this definition
agrees with the old one. Also note that we allow the functions to take infinite value here.
3. Suppose {f } is a sequence of non-negative measurable functions defined on E and {f }
n n
converges (pointwisely) to a .................................. f a.e. on E. Then f £ lim inf f .
E ò
E ò
n n
4. If {f } is an ..................................... of non-negative measurable functions defined on E
n
(increasing in the sense that f £ f for all n on E) and f  f a.e. on E, then ò f ­ f by
n n+1 n E n E ò
which it means {j f } is an increasing sequence with limit ò f .
E n E
5. A ................................. f defined on E is integrable if and only if |f|<  so.
E ò
6. For any f,g Î (E) and A  E, we have ò (f g) = A ò f + A ò g and ò  =  f . Furthermore,
+
f
A A A ò
if f £ g a.e. on A then ................................... .
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