Bounded Continuous Function Open Interval is Lebesgue Integrable

7. The Integral

7.4. Lebesgue Integral

We previously defined the Riemann integral roughly as follows:

• subdivide the domain of the function (usually a closed, bounded interval) into finitely many subintervals (the partition)
• construct a simple function that has a constant value on each of the subintervals of the partition (the Upper and Lower sums)
• take the limit of these simple functions as you add more and more points to the partition.

If the limit exists it is called the Riemann integral and the function is called Riemann integrable. Now we will take, in a manner of speaking, the "opposite" approach:

• subdivide the range of the function into finitely many pieces
• construct a simple function by taking a function whose values are those finitely many numbers
• take the limit of these simple functions as you add more and more points in the range of the original function

If the limit exists it is called the Lebesgue integral and the function is called Lebesgue integrable. To define this new concept we use several steps:

1. we define the Lebesgue Integral for "simple functions"
2. we define the Lebesgue integral for bounded functions over sets of finite measure
3. we define the general Lebesgue integral for measurable functions

First, we need to clarify what we mean by "simple function".

A function f defined on a measurable set A that takes no more than finitely many distinct values a₁, a₂, ... , a_n can always be written as a simple function

f(x) = a_n X_An (x)

where

A _n = { x A : f(x) = a_n }

Therefore simple functions can be thought of as dividing the range of f, where the resulting sets A _n may or may not be intervals.

Examples 7.4.2: Simple Functions

A step function is a function s(x) such that s(x) = cj for xj-1 < x < xj and the { xj } form a partition of [a, b]. Upper, Lower, and Riemann sums are examples of step functions. What is the difference, if any, between step functions and simple functions. Are simple functions uniquely determined? In other words, if s1 and s2 are two simple functions with s1(x) = s2(x), do they have to have the same representation? If different representations are possible, which one is "the best"? How does the Dirichlet function fit in with this terminology? Is the function that is equal to 1 if x is part of the Cantor middle-third set and 0 otherwise a simple function? Are sums, differences, and products of simple functions simple?

For simple functions we define the Lebesgue integral as follows:

Just as step functions were used to define the Riemann integral of a bounded function f over an interval [a, b], simple functions are used to define the Lebesgue integral of f over a set of finite measure.

Now that we have defined the Lebesgue integral for bounded functions, we want to know exactly what bounded functions are in fact Lebesgue integrable.

Theorem 7.4.7: Lebesgue Integrable Bounded Functions
	If f is a bounded function defined on a measurable set E with finite measure. Then f is integrable if the sets Ek = { x E: (k-1) M / n ≤ f(x) < k M/n } for k = -n, -(n-1), ... 0, 1, 2, ..., (n-1), n are measurable, then f is Lebesgue integrable. Proof

Now a function f can be integrated (if it is integrable) using either the Riemann or the Lebesgue integral. Fortunately, for many simple functions the two integrals agree and the Lebesgue integral is indeed a generalization of the Riemann integral.

For most practial applications this theorem is all that is necessary: for continuous functions or bounded functions with at most countably many discontinuities over intervals [a, b] there is no need to distinguish between the Lebesgue or Riemann integral. All integration techniques we learned apply equally well, using either integral. But for more complicated situations or more theoretical purposes the Lebesgue integral is more useful, but then techniques such as integration by parts or substitution may no longer apply.

The Lebesgue integral has properties similar to those of the Riemann integral, but it is "more forgiving": you can change a function on a set of measure zero without changing the integral at all.

At this point we could stop: we have extended the concept of integration to (bounded) functions defined on general sets (measurable sets with finite measure) without using partitions (subintervals). The new concept, the Lebesgue integral, agrees with the old one, Riemann integral, when both apply, and it removes some of the oddities mentioned before.

But as the astute reader has surely already noticed, we have restricted our definition of Lebesgue integrable function to bounded functions only. Can we continue to generalize the Lebesgue integral to functions that are unbounded, including functions that may occasionally be equal to infinity? To do that, we first need to define the concept of a measurable function.

In other words, functions whose values are real numbers or possibly plus or minus infinity are measurable if the inverse image of every interval (-, a) is measurable.

That is somewhat comparable to one of the equivalent definitions of continuous functions: a function f is continuous if the inverse image of every open interval is open. However, not every measurable function is continuous, while every continuous function is clearly measurable.

Measurable functions that are bounded are equivalent to Lebesgue integrable functions.

Measurable functions do not have to be continuous, they may be unbounded and they can, in particular, be equal to plus or minus infinity. On the other hand, measurable functions are "almost" continous.

Using measurable functions allows us to extend the Lebesgue integral first to non-negative functions that are not necessarily bounded and then to general measurable functions.

The final step to define the Lebesgue integral of a general function is now easy.

Proposition 7.4.X remains true for general Lebegues integrable functions.

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Source: https://mathcs.org/analysis/reals/integ/lebes.html

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