# Measure Theory/Properties of Bounded Integrals
## Basic Properties of Bounded Integrals, and the Bounded Convergence Theorem

In this section we prove what I term the "basic properties of integrals".
After proving basic properties, we prove the first of our "swaparoo" theorems: the bounded convergence theorem.
Throughout this lesson you may assume
* $E\in {\mathcal {M}},\quad \lambda (E)<\infty$
* $f,g,f_{n}:E\to {\mathbb {R}}$ are bounded, measurable functions for all $1\leq n$
* $c,d\in {\mathbb {R}}$
* $f_{n}{\overset {p.w.}{\to }}f$

### Linearity
#### Exercise 1. Bounded Integral Linearity

Prove $\int _{E}(f+g)=\int _{E}f+\int _{E}g$.
Hint: Often with problems involving a sup on one side and a sum of sups on the other, in abstract like $\sup A=\sup B+\sup C$, it can be easiest to show that the right-hand side is the least upper bound of A. (I would muse that this is perhaps because the left-hand side is a single sup and therefore is asymmetrically more tractable than the other sups.)
Anyway, to get started, let $\varphi :E\to {\mathbb {R}}$ be a simple function with $\varphi \leq f+g$. Show that $\int _{E}\varphi \leq \int _{E}f+\int _{E}g$. (Insofar as we are trying to show that $\int _{E}f+\int _{E}g$ meets the definition of the "least upper bound", which part does this demonstrate?)
Next let $\varepsilon \in {\mathbb {R}}^{+}$ and show that $\int _{E}f+\int _{E}g-\varepsilon$ is not an upper bound on the set of all $\int _{E}\varphi$ for simple functions $\varphi \leq f+g$. Hint: $\int _{E}f+\int _{E}g-\varepsilon =\left(\int _{E}f-\varepsilon /2\right)+\left(\int _{E}g-\varepsilon /2\right)$.
Next prove that $\int _{E}cf=c\int _{E}f$. Hint: consider cases depending on whether c is positive, negative, or zero. Moving a constant multiple out of a supremum behaves differently depending on the case.
Finally, infer $\int _{E}(cf+dg)=c\int _{E}f+d\int _{E}g$.

### Inequality Preservation

Suppose $f\leq g$.

#### Exercise 2. Prove Inequality Preservation

Prove that $\int f\leq \int g$.

### The "Integral Triangle Inequality"

Nobody that I have ever seen uses this name, but it makes a lot of sense to me. Here is what I am calling the "integral triangle inequality":
    $\left|\int _{E}f\right|\leq \int _{E}|f|$
Note that this name makes sense since an integral is like a "smooth sum" and the triangle inequality for sums is $\left|\sum a_{n}\right|\leq \sum |a_{n}|$. (I omit details like the indexing set and nature of the sequence, for simplicity and presumably the reader can fill them in however one likes.)

#### Exercise 3. Prove the Integral Triangle Inequality

Use the previous exercise's result to show
    $-\int _{E}|f|\leq \int _{E}f\leq \int _{E}|f|$

### The ML Bound

Again I am using a slightly strange name for the result -- this time it is strange only because the name "ML bound" is typically only used in the context of complex analysis and multivariable calculus. But the idea is the same: If f is bounded by M on a "curve" (for us, because we are integrating in one dimension, a "curve" would just stand in analogy to a measurable set) of length L, then the integral is bounded by ML. For us in this setting, $L=\lambda (E)$.

#### Exercise 4. Prove the ML Bound

Let f be bounded by M, which is to say $|f|\leq M$. Also assume $\lambda (E)=L$.
Prove that
    $\int _{E}f\leq ML$
Hint: Use two earlier results.

### Integral Finite Additivity

This is the property that, if $E=A\sqcup B$ for some two measurable subsets $A,B\subseteq E$, then
    $\int _{E}f=\int _{A}f+\int _{B}f$
At least, this is the "pairwise" version of finite additivity. We could generalize this to any decomposition of E into any finite partition.

#### Exercise 5. Prove Integral Finite Additivity

Prove integral pairwise additivity, by applying linearity to $\int _{E}f=\int _{E}f\cdot (\mathbf {1} _{A}+\mathbf {1} _{B})$.
Then state the generalization to any partition of E into a finite number of cells, and then prove the generalization.

### The Bounded Convergence Theorem

A very big moment in your life is coming up, if you are new to measure theory! You are about to get your first convergence theorem, which I have been cutely calling "swaparoo".
The Bounded Convergence Theorem is not the most celebrated convergence theorem -- that one is probably the Monotone Convergence Theorem (MCT), because of how important it has proved for various applications. Still, it is a big deal in its own right, and will help us to prove the MCT.
Recall $f_{n}$ defined above, and assume moreover that there is a single common bound M for all $f_{n}$. That is to say,
    $\forall x\in E,\ \forall n\in {\mathbb {Z}}^{+},\quad |f_{n}(x)|\leq M$

#### Exercise 5. Prove the BCT

Prove that
    $\lim _{n\to \infty }\int _{E}f_{n}=\int _{E}\lim _{n\to \infty }f_{n}(x)=\int _{E}f$
Hint: There is a clue about what might be the right earlier result to use here. This is about a sequence of measurable functions.
Show that for each $\varepsilon \in {\mathbb {R}}^{+}$, there is an $n\in {\mathbb {Z}}^{+}$ such that
    $\left|\int _{E}f_{n}-\int _{E}f\right|<\varepsilon$
by using: linearity, triangle inequality, split.