Measure Theory/Length-integration Defined

Length-integration Defined
Now that we have a good understanding of measurable functions, we are in a position to formally define the length-measure integral. Still this must be built up in stages.

To begin with we define characteristic functions, then simple functions, and then the integral of simple functions.

We can then take a nonnegative function, f, and approximate it by simple functions. We then use the integral of the simple functions, to approximate the integral of f.

Exercise 1. Simple Simple Function Example
Consider the function $$f(x) = x^2$$ on the interval [0,2], and the simple function $$psi = \mathbf 1_{[1,1.5]}+(1.5^2)\mathbf 1_{(1.5,2]}$$.

Graph both functions and prove that $$\psi\le f$$.

Exercise 2. Make It Simple
The function $$\psi=\mathbf 1_{[0,1]}+\mathbf 1_{[0,2]}$$ doesn't look like a simple function because the characteristic functions are not disjoint (or, rather, their sets are not disjoint). However, it is a simple function.

Write the function $$\psi$$ in the form of a characteristic function. Hint: It takes a constant value on [0,1] and a different value on the set (1,2].

Exercise 3. Simple Functions Have Canonical Form
Prove that every function of the form $$\sum_{i=1}^n c_i\mathbf 1_{E_i}$$, for real numbers $$c_1,\dots,c_n$$ and measurable sets $$E_1,\dots,E_n$$, is a simple function which has a canonical form.

Hint: First prove that one can always replace the terms of $$\sum_{i=1}^n c_i\mathbf 1_{E_i}$$ with new terms such that the coefficients are all distinct. Do so by considering any two terms, $$c_i\mathbf 1_{E_i}$$ and $$c_j\mathbf 1_{E_j}$$, with $$i\ne j$$ but $$c_i=c_j$$, and showing that these can be condensed into a single equivalent term.

Next prove that, if all the coefficients are distinct in $$\sum_{i=1}^n c_i\mathbf 1_{E_i}$$, then for any overlapping sets $$E_i,E_j$$ with $$i\ne j$$, the terms $$c_i\mathbf 1_{E_i},c_j\mathbf 1_{E_j}$$ may be replaced by three terms which have mutually disjoint characterized sets.

Finally, prove that if one iteratively applies the following algorithm, the result will be a simple function in canonical form:

1. Take the term $$c_1\mathbf 1_{E_1}$$ and pairwise apply the above result to all other terms, resulting in $$\sum_{i=1}^{n_1} c_{i,1}\mathbf 1_{E_{i,1}}$$.

2. Take the term $$c_2\mathbf 1_{E_2}$$ and pairwise apply the above result to all other terms in $$\sum_{i=1}^{n_1} c_{i,1}\mathbf 1_{E_{i,1}}$$, resulting in $$\sum_{i=1}^{n_2} c_{i,2} \mathbf 1_{E_{i,2}}$$.

3. Proceed likewise until, and including, the term $$c_n\mathbf 1_{E_n}$$.

Exercise 4. Sums of Simple Functions
Let $$\varphi=\sum_{i=1}^m c_i\mathbf 1_{E_i}, \quad \psi=\sum_{i=1}^n d_i\mathbf 1_{F_i}$$ be two simple functions in canonical form.

Add to the collection of sets $$E_1,\dots, E_m$$ the set $$E_0$$ which is the set of points on which $$\varphi(x) = 0$$. Also add to $$F_1,\dots,F_n$$ the set $$F_0$$ where $$\psi(x)=0$$.

Now let $$G_k$$ be all the sets which result from any intersection $$E_i\cap F_j$$ for $$0\le i\le m,\ 0\le j\le n$$. Also, if $$1\le k\le (m+1)(n+1)$$ and $$G_k=E_i\cap F_j$$, then define $$c'_k = \begin{cases}0 & \text{ if } i=0 \\ c_i & \text{ if } 1\le i\le m\end{cases}$$. Likewise define $$d'_k=\begin{cases} 0 & \text{ if } j=0\\ d_j & \text{ if } 1\le j\le n\end{cases}$$.

Now show that $$\varphi = \sum_{k=1}^{(m+1)(n+1)} c'_k \mathbf 1_{G_k}$$, and likewise for $$\psi$$.

Infer that $$\varphi+\psi = \sum_{k=1}^{(m+1)(n+1)} (c'_k+d'_k)\mathbf 1_{G_k}$$.

Exercise 4. Simple Measurable
Prove that every simple function is measurable.

Integral of a Simple Function
Notice a few things about this. For one thing, integrating a simple function turns a $$\mathbf 1$$ into a $$\lambda$$. This helps one to see that the integral of the simple function is a number, because it is a linear combination of numbers.

Also you probably noticed the lack of a differential. Some authors will write the integral as $$\int \psi\ d\lambda$$ to express that the integral is taken "with respect to length-measure". However, it is also common to simply drop the differential, since everything we see for many sections to come will all use the length-measure.

Exercise 5. Simple Integral
Compute $$\int\left(\chi_{[0,1]}+2\chi_{[0,2]}\right)$$.

Integral of a Bounded Function
Finally, that elusive definition is at hand!

Integral of a Nonnegative Function
In the following definition we relax the conditions of boundedness, both on E and f. However, we impose the condition of nonnegativity.

Notice that in each definition of integration, it is built on the back of the previous definition.

General Length-measure Integral
Finally, we remove all conditions and obtain the most general definition of integration that we will discuss in this course.

Recall that $$f^+,f^-$$ refer to the positive and negative parts of a function, as defined in a previous lesson. We proved that if f is measurable then so are the parts, and therefore these integrals are well-defined.

By the way, you may notice the lack of bounds of integration. All integrals are assumed to take place over the entire domain of f. If we ever need to restrict an integral to a subset $$F\subseteq E$$, then we may do so by instead integrating $$\int_E f\mathbf 1_{F}$$.

In fact, this is the same as merely restricting f to F and then integrating $$\int_F f$$. We will prove this claim later when we have more tools to do so.