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The Lebesgue integral. Theorems. Bounded, dominated, monotone convergence theorems.

Lebesgue integral. Let f(x)
be a bounded measurable
function defined over a
(Lebesgue) measurable set E of
finite measure (for intuitive
insight view f(x) as the function
y = f(x) shown in Fig. 1 defined
on the interval [a, b] --- where E
corresponds to the interval [a,
b]). Choose two real numbers A
and B such that the range of y =
f(x) lies between A and B [A
and B represent lower and upper
bounds of f(x)]. Divide the
interval of the y axis from A to
B up into n subintervals by
choosing points y_{0} = A, y_{1}, y_{2}, .... , y_{n} = B as shown in Fig. 1. Let

i.e. *E*_{i} is the subset of E consisting of the set of x ε E for which

See Fig. 1.

An approximation to the Lebesgue integral of the function f(x) is the Lebesgue integral sum

1) S = y_{1 }· mE_{1} + y_{2 }· mE_{2} + .... + y_{i }· mE_{i} + ..... y_{n }· mE_{n}

where mE_{i} is the measure of point set E_{i}. The Lebesgue integral

is the limit of the Lebesgue integral sum S when max |y_{i -1 }- y_{i}| → 0 and n → ∞.

Note. In place of 1) the sum

2) S = y_{0 }· mE_{1} + y_{1 }· mE_{2} + .... + y_{i -1 }· mE_{i} + ..... y_{n -1 }· mE_{n}

can also be used. Both give the same result.

Theorems on Lebesgue integrals of bounded functions

In the following we assume that all sets are measurable and of finite measure and that f(x) is bounded and measurable and thus Lebesgue integrable.

1. For any constant c

2. For any constant c

3. If E has measure zero, then

4. Mean-value theorem. If A f(x) B, then

5. If E = E_{1} ∪_{ }E_{2} where E_{1} and E_{2} are disjoint, then

6. If E = E_{1 }∪_{ }E_{2} ∪ ... where E_{1}, E_{2}, ..... are mutually disjoint, then

8. If f(x) and g(x) are bounded and measurable on E, then f(x)g(x) is Lebesgue integrable on E i.e.

9. If f(x) g(x) on E, or almost everywhere on E, then

10. If f(x) is bounded and measurable on E, then |f(x)| is Lebesgue integrable on E. Conversely, if |f(x)| is bounded and measurable on E, then f(x) is Lebesgue integrable on E.

11. If f(x) is bounded and measurable on E, then

12. If f(x) = g(x) almost everywhere on E, then

13. If f(x) 0 almost everywhere on E and

then f(x) = 0 almost everywhere on E.

The Lebesgue integral has one remarkable property that the Riemann integral does not have. It is the property given by the following theorem.

Bounded convergence theorem. Let {f_{n}(x)} be a sequence of measurable functions
defined on an interval [a, b] that converges almost everywhere to f(x). If these functions are
uniformly bounded i.e. there exists a constant K such that

|f_{n}(x)| < K

for every n and every x in [a, b], then

Bounded convergence theorem for infinite series. Let u_{1}(x), u_{2}(x), .... be measurable
on [a, b] and the partial sums

be uniformly bounded on E (i.e. there exists a constant K such that |s_{n}(x)| < K for every n and all
x ε [a, b] ) and

Then

Relationship between Riemann and Lebesgue integrals. If f(x) is Riemann integrable in [a, b], then it is Lebesgue integrable in [a, b] and the two integrals are equal. The converse is, however, not true. If f(x) is Lebesgue integrable in [a, b], it need not be Riemann integrable in [a, b].

Theorem 1. A function is Riemann integrable in [a, b] if and only if the set of discontinuities of f(x) in [a, b] has measure zero i.e. if f(x) is continuous almost everywhere.

Theorem 2. If f(x) is continuous almost everywhere in [a, b], then it is Lebesgue integrable in [a, b].

*********************************************************************

Def. Lebesgue integral for unbounded functions. Let f(x) be an unbounded measurable function defined over a measurable set E of finite measure. Define as follows:

Then f(x) has the Lebesgue integral

provided this limit exists.

For intuitive insight see Fig. 2a and 2b for a function f(x) defined on the interval [0, 10].

If the set E does not have finite measure and

approaches a limit as the boundaries of an interval I all increase indefinitely, in any manner, then that limit is defined as

Theorems on Lebesgue integrals of bounded functions

In the following we assume that all sets and functions are measurable.

1. If f(x) 0, then

exists if and only if

is uniformly bounded.

2. If |f(x)| g(x) almost everywhere on E and g(x) is integrable on E, then f(x) is also integrable on E and

3. A function f(x) is integrable on E if and only if |f(x)| is integrable on E and in such case

Because of this we say that f(x) is integrable on E if and only if it is absolutely integrable on E.

4. If

exists, then f(x) is finite almost everywhere in E.

5. If E has measure zero, then

6. If

exists and if A is a measurable subset of E, then

also exists. In such case we have

7. Let E = E_{1 }∪ E_{2 }∪_{ ... } where E_{1}, E_{2}, ..... are mutually disjoint. Then if

exists

9. For any constant c

10. If f(x) is integrable on E and g(x) is bounded, then f(x)g(x) is integrable on E.

11. If f(x) = g(x) almost everywhere on E, then

12. If f(x) 0 almost everywhere on E and

then f(x) = 0 almost everywhere on E.

13. If f(x) is integrable on E, then given ε > 0 there exist a δ > 0 and a set A E such that if mA < δ

14. Let f(x) be integrable in E. If {E_{k}}is a sequence of sets contained in E such that
= 0,
then

Source: Spiegel. Real Variables (Schaum)

Dominated convergence theorem. Let {f_{n}(x)} be a sequence of measurable functions
defined on an interval [a, b] that converges almost everywhere to f(x). Then if there exists a
function M(x) integrable on E such that

|f_{n}(x)|
M(x)

for every n, then

Dominated convergence theorem for infinite series. Let u_{1}(x), u_{2}(x), .... be
measurable on [a, b]. Let there exist an integrable function M(x) on [a, b] such that |s_{n}(x)| ≤
M(x) where s_{n}(x) is the partial sum

and let

Then

Theorem. If for the series

the condition

holds for some constant M and if v(x) is bounded and measurable on [a, b], then

Fatou’s theorem. Let {f_{n}(x)} be a sequence of non-negative measurable functions defined on
[a, b] and suppose that the sequence converges to f(x) almost everywhere. Then

Monotone convergence theorem. Let {f_{n}(x)} be a sequence of non-negative monotonic
increasing functions defined on [a, b] and suppose that the sequence converges to f(x). Then

Theorem. Let u_{k}(x)
0, k = 1, 2, .... . Then

provided either side converges.

References

James and James. Mathematics Dictionary

Spiegel. Real Variables (Schaum)

Mathematics, Its Content, Methods and Meaning.

Natanson. Theory of Functions of a Real Variable

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