Terminology: we say the limit of a sequence or sum, or Sup or Inf of a set exists if it is finite (and unique). If it is or we say the limit/Sup/Inf is defined, i.e. unbounded, but does not exist. In any other cases, e.g , we say it does not exist (or undefined).
Definition: A partition of a set is a collection of non-empty (pairwise) disjoint subsets of whose union equals .
Definition: A real-valued simple function is a function that takes a finite or countable number of real values (NOT extended real values including ), i.e. its range is finite or countable. Note that the definition does not put any restriction on the domain or codomain of the function.
For example defined as,
is a simple function.
A simple function on an interval can be defined as,
where are constants, such that and for , and is called the characteristic function of being defined as,
In words, the interval was partitioned into pairwise disjoint intervals s where .
The above definition is called the canonical representation of a simple function and is equivalent to,
The domain of a simple function is not restricted to ; a simple function can be with its domain being any set.
Remark: a step function is a type of a simple function, i.e. a simple function is called a step function when and each is an interval of real numbers.
Definition: a measurable function is a simple measurable function if it is a simple function.
Approximating a function by simple function(s)
Let and is the range of the function. If we partition the range as where the subsets are pairwise disjoint, we can construct a simple function approximating as,
where , and . Note that the is a partition of disjoint sets because s are pairwise disjoint. Also, .
To construct the simple function approximation, the range of the function was partitioned and then the domain was implicitly partitioned through the pre image. The reason is that firstly partitioning the real line as the codomain/range can be readly performed. Secondly, this approach is the foundation of Lebesgue integration. Lebesgue integration partitions the range into intervals, then each summand is a number in one of the intervals in the partition times the measure of the preimage of that interval. This makes Lebesgue integration capable of considering functions that are not Riemann integrable and also having nice properties like interchanging limit and integration operators. Any real-valued function can be written as the limit of a sequence of simple functions. For non-negative functions, however, the following theorem exists.
Theorem L1: Let be a non-negative measurable function. Then, there exists a monotonically increasing sequence of non-negeative measurable simple functions (where ) such that (this is a pointwise convergence). The approximation of by is an approximation from bellow, i.e. . Such a sequence can be constructed as (this proves the existence),
This is explained as follows. Regarding each , the codomain of is partitioned as . Then, is devided/partitioned into subintervals such that , i.e.,
The rest of the codomain is considered as regarding an .Then, the preimage () of each subinterval is determined (on the domain) as, and . These preimages partition the domain into pairwise disjoint sets because the partitioning sets of the codomain are disjoint. With these assumptions, each is a simple function. Note that a simple function is bounded.
Proposition L1: Any real-valued function can be approximated by monotonically increasing sequence of non-negeative measurable simple functions if written as where, and . Note that both and are positive and hence satisfies the conditions of Theorem L1.
Proposition L2: if is bounded, then converges uniformely.
Integration with respect to a measure/ Lebesgue integral
Definition: Let be a measure space and be a bounded measurable function. If , and is a disjoint measurable partition of , i.e. and , define,
Also, if is another disjoint partition of , we write and say is a refinement of if for some .
Then, we can show,
- i.e. increasing sequence.
- i.e. decreasing sequence.
- . i.e. as refines.
We say is Lebesgue integrable over if . In this case we write,
Remark: can be a set of any objects (numbers, symbols, animals, etc).
The Lebesgue integration of a simple measurable function over a set of finite measure
Theorem L2: Let be a measure space and as and a measurable function where , and s are pairwise disjoint. Then, is Lebesgue integrable on iff exists. In this case we write,
For the proof, note that is a (disjoint and measurable) partition of and on each .
In above formulation, the sum is over the size of the partition, i.e and s are sequentially in correspondence with s. We can however write the sum over the size of the range of the simple function. The range of a simple measurable function is a discrete, i.e. countable set, . If the number of elements in (the size of) is and , then,
Note that the range of the function is a set and it indeed contains distict values.
Example: Let be the Lebesgue measure and be defined as,
Then, Lebesgue integration by summing over the size of the partition gives,
And, the integration by summing over the range of the function () gives,
Example: Let be a measure space and,
if , we calculate as follows.
We note that the support of the integral has a finite measure with respect to the defined measure; because, . Therefore, we can write,
Remark: Lebesgue integrability depends on the measure envolved.
Remark: if then for any set , we can write
Proposition [L3]: The follwing properties hold for integration of simple measurble functions over sets of finte measure:
- If then .
- If on , then .
- If almost every where on , then .
The Lebesgue integration of a bounded measurable function over a set of finite measure
The general definition of Lebesgue integration and the formulation for the Lebesgue integration of a simple function can be used to (re-) define the definition of the Lebesgue integration of a bounded function on a set of finite measure. In this regard, the lower and upper Lebesgue integrals of on with are defined as,
which are bounded and .
in which is a measure on ; the operations Sup and Inf are over all simple functions approximating the function.
By definition, if , we say is Lebesgue integrable over and its integral equals the common value and is denoted as .
Theorem [Lebesgue integribility of functions]: If is a bounded measurable function on a set of finite measure, where , then is Lebesgue integrable, i.e. exists. The converse is also true, stating that, if a bounded function on a set of finite measure is integrable (i.e. exists), then the function is a measurable function.
Proposition L4: Proposition L3 holds for the above afformentioned functions.
Theorem L3: Let be a bounded measurable function on with . If where is at most a countable family of pairwise disjoint measurable sets, then .
An example that can be solved by the above theorem is such that where , and such that and otherwise .
Proposition [Lebesgue and Riemann integrations]: Let be a bounded measurable function where in the Lebesgue measure (n-dimensional interval length). If has a finite number of discountinuity and and for . Then, the following integral exists and,
Lebesgue integration: = Riemann integration:
Riemann integration of a bounded function on a set of finite measure can be regarded as a particular case of the general Lebesgue integration. In fact, Riemann integration is based on subdividing the domain of a function whereas Lebesgue integration is based on subdividing the range of a function and using the inverse image to create measurable subdivision on the domain of the function.
The Lebesgue integration of an unbounded measurable function over a measurable set
Riemann integration is defined as the limit of the Riemann sum for bounded functions on bounded domains. For unbounded functions or domains, the Riemann integration is defined as limits:
1- If is continuous (with finite number of discountinuity though) on and unbounded at as , then the follwing Riemann integral is defined provided that the limit on the RHS exists (i.e. being finite),
that is calculating the integral for a bounded function and then taking the limit.
If is unbounded as , we define . And if is unbounded at , then we split the integral at and write the limits.
2- If the domain of integration is unbounded with respect to a variable, then the integration is defined as a limit if exists. For example, if continuous, then,
The above integrations are classified as improper Riemann integrations. Lebesgue integration approaches these unbounded cases in a natural (more general) way. Therefore, the temr improper is not used with these cases of Lebesgue integration.
Definition: A measurable function on a set is of finite support if there is a set for which and on . The support of then becomes the set over which the function does not vanish.
Proposition: Let be a bounded measurable function on with a finite support , then .
Proof: by assumption on . The proposition is already proved If . For the case that , i.e. unbounded, we can write,
Therefore, we can write as a simple function,
by using the formula for Lebesgue integration of a simple function we conclude that,
Remark: in above we showed that even if . Note that we did not write ; this expressin is undefined. But, we used being equal to zero. In other words, one may write, which is undefined. We should note that the theorem on the integration of simple function holds for sets of finite measure.
Proposition: Let be a bounded measurable function on (with finite or infinite measure). If almost everywhere on , then .
To move ahead and define Lebesgue integration for any measurable functions including unbounded ons on any measurable support, non-negative functions are considered first. Considering allows using lower approximations of the function by bounded functions of finite support.
Definition L1 [Lebesgue integration of non-negative functions]: For on , the integration is defined as,
the suprimum is on all functions described as . If the suprimum of the above set of values exists, we say the function is integrable. If value is infnity the integral is defined and is unbounded. Note that is a pointwise expression means either or at each . Also, each is of finite support, meaning that except over some part of the domain with a finite measure, valishes over te rest of its domain.
Above definition can also prove if almost everywhere on , then . To this end let .
Definition L2 [Lebesgue integration of functions]: The Lebesgue integration for any measurable function and a measurable set is defined as,
provided that at least one of the integrals is finite. If the integral equals or the integral is defined however we say the function is integrable if the integral is finite. The case is undefined.
Theorem L4: let be a non-negative simple function and . Then, .
Note that since the sum is always defined, either finite or .
Theorem L5: Let be non-negative. Define for all . Then
(a) is countably additive on . I.e. for and as . Which means,
(b) for any not necessarily positive function, is countably additive on if is integrable on .
(a) If is a simple function such that , by theorem L4, we can write, . Therefore, by definition L1,
Because and is definitely a non-negative (countably additive) measure, it is clear that for any . Note that . Now, If for some , then and by the above results , therefore, and it is trivial.
So, we assume that , i.e finite. Therefore, for any we can let and find a simple function as such that,
Because is arbitrary, above indicates that . It follows that (by induction if you want), for any ,
And because ,
Therefore (considering the first part of the proof),
(b) If is integrable, then each and exists and the proof of (a) can be applied to each part.
Corollary L1: for sets and such that and , then . This shows that a set of measure zero is negligible in integration.
Proposition: Let be a non-negative bounded measurable function on . If , then almost everywhere on .
Theorem L6: If a measurable function is (Lebesgue) integrable (finite in fact) with respect to a measure on , then is (Lebesgue) integrable on , and .
Proof: Let as a disjoint partition such that and . Then, by theorem L5,
For the second part, since we can write and which implies .
By the above theorem, we see that integribility of implies that of . Because of that, Lebesgue integral is called absolutely convergent integral. It should be noted that Riemann integration is not neccessarily absolutely convergent.
Theorem L7: For a measurable function on , if and is integrable on . Then is integrable on .
Theorem L8 [Lebesgue’s monotone convergence theorem]: For a measurable set , let be a sequence of measurable functions, , such that
Let be defined as the following pointwise convergence,
Note that the sequence converges to from below. Also may or may not be bounded and hence the integral. for proof see [WR].
Theorem L8: For a measurable set , if is a sequence of nonnegative measurable functions () and
Note that has to be nonnegative but doesn’t need to be monotone. Proof of this theorem is by noting that the partial sums of the infinite sum form a monotonically increasing sequence and using Theorem L7.
Theorem L9 [Fatou’s theorem]: For a measurable set if is a sequence of nonnegative measurable functions and , then
Note that does not need to be monotone.
From theorem L9, we can conclude that if a measurable function is a limit of a sequence of nonnegative measurable functions , then . This is because for any sequence of functions . If the sequence is monotonically increasing functions then Theorem L8 holds.
Theorem L10 [Lebesgue’s dominated convergence theorem]: for a measurable set , let be a sequence of measurable functions such that . If there exists a measurable function on such that for all and , meaning that is uniformly bounded, then
Corollary L2: If , i.e. finite measure, and is uniformly bounded on , and on , then theorem L10 holds.
[WR] Walter Rudin-Principles of Mathematical Analysis, Third Edition-McGraw-Hill Science Engineering Math (1976)