Measure Theory/Integration

Let (X,σ,μ) be a σ-finite measure space. Suppose s is a positive simple measurable function, with $s=\displaystyle\sum_{i=1}^3y_i\chi_{A_i}$; $A_i\in\sigma$ are disjoint.

Define $\displaystyle\int_Xs~d\mu=\sum y_i\mu(A_i)$

Let $f:X\to\overline{\mathbb{R}}$ be measurable, and let $f\geq 0$.

Define $\displaystyle\int_Xf~d\mu=\sup\{\int_Xs~d\mu=\sum y_i\mu(A_i)|s\text{ simple },s\geq0,s\leq f\}$

Now let f be any measurable function. We say that f is integrable if f ⁺ and f ⁻ are integrable and if $\displaystyle\int_Xf^+~d\mu,\int_Xf^-~d\mu<\infty$. Then, we write

$\displaystyle\int_Xf~d\mu=\int_Xf^+~d\mu-\int_Xf^-~d\mu$

The class of measurable functions on X is denoted by $\mathcal{L}^1(X)$

For $0<p<\infty$, we define $\mathcal{L}^p$ to be the collection of all measurable functions f such that $|f|^p\in\mathcal{L}^1$

A property is said to hold almost everywhere if the set of all points where the property does not hold has measure zero.

Properties

Let (X,σ,μ) be a measure space and let f,g be measurable on X. Then

1. If $f\leq g$, then $\displaystyle\int_Xfd\mu\leq\int_Xgd\mu$
2. If $A,B\in\sigma$, $A\subset B$, then $\displaystyle\int_Afd\mu\leq\int_Bfd\mu$
3. If $f\geq 0$ and $c\geq 0$ then $\displaystyle\int_Xcfd\mu=c\int_Xfd\mu$
4. If $E\in\sigma$, μ(E) = 0, then $\displaystyle\int_Efd\mu=0$, even if $f(E)=\{\infty\}$
5. If $E\in\sigma$, f(E) = {0}, then $\displaystyle\int_Efd\mu=0$, even if $\mu(E)=\infty$

Proof

Monotone Convergence Theorem

Suppose $f_n\geq 0$ and f_n are measurable for all n such that

1. $f_1(x)\leq f_2(x)\leq\ldots$ for every $x\in X$
2. $f_n(x)\to f(x)$ almost everywhere on X

Then, $\displaystyle\int_Xf_nd\mu\to\int_Xfd\mu$

Proof

$\displaystyle\int_Xf_nd\mu$ is an increasing sequence in $\mathbb{R}$, and hence, $\displaystyle\int_Xf_nd\mu\to\alpha\in\overline{\mathbb{R}}$ (say). We know that f is measurable and that $f\geq f_n\forall n$. That is,

$\displaystyle\int_Xf_1d\mu\leq\int_Xf_2d\mu\leq\ldots\int_Xf_nd\mu\leq\ldots\int_Xfd\mu$

Hence, $\displaystyle\alpha\leq\int_Xfd\mu=\sup\{\int_Xsd\mu:s\text{ is simple },0\leq s\leq1\}$

Let $c\in [0,1]$

Define $E_n=\{x|f_n(x)\geq cs(x)\}$; $n=1,2\ldots$. Observe that $E_1\subset E_2\subset\ldots$ and $\bigcup E_n=X$

Suppose $x\in X$. If f(x) = 0 then s(x) = 0 implying that $x\in E_1$. If f(x) > 0, then there exists n such that f_n(x) > cs(x) and hence, $x\in E_n$.

Thus, $\bigcup E_n=X$, therefore $\displaystyle\int_Xf_n(x)d\mu\geq\int_{E_n}f_nd\mu\geq c\int_{E_n}sd\mu$. As this is true if $c\in [0,1]$, we have that $\alpha\geq\displaystyle\int_Xfd\mu$. Thus, $\displaystyle\int_Xf_nd\mu\to\int_Xfd\mu$.

Fatou's Lemma

Let $f_n\geq 0$ be measurable functions. Then,

$\displaystyle\int_X\liminf f_n d\mu\leq\liminf\int_Xf_nd\mu$

Proof

For $k=1,2,\ldots$ define $g_k(x)=\displaystyle\inf_{i\geq k}f_i(x)$. Observe that g_k are measurable and increasing for all x.

As $k\to\infty$, $g_k(x)\to\displaystyle\liminf_{n\geq k}f_n(x)$. By monotone convergence theorem,

$\displaystyle\int_Xg_kd\mu\to\int_X\liminf f_n(x)d\mu$ and as $\displaystyle\int_Xg_k(x)d\mu\leq\liminf\int g_k(x)d\mu$, we have the result.

Dominated convergence theorem

Let (X,σ,μ) be a complex measure space. Let {f_n} be a sequence of complex measurable functions that converge pointwise to f; $f(x)=\displaystyle\lim_{n\to\infty}f_n(x)$, with $x\in X$

Suppose $|f_n(x)|\leq g(x)$ for some $g\in\mathcal{L}^1(X)$ then

$f\in\mathcal{L}^1$ and $\displaystyle\int_X|f_n-f|d\mu\to 0$ as $n\to\infty$

Proof

We know that $|f|\leq g$ and hence $|f_n-f|\leq 2g$, that is, $0\leq 2g-|f_n-f|$

Therefore, by Fatou's lemma, $\displaystyle\int_X2gd\mu\leq\liminf\int_X(2g-|f_n-f|)d\mu\leq\displaystyle\int_X2gd\mu+\liminf\int_X(-|f_n-f|)d\mu$

$=\displaystyle\int_X2gd\mu-\limsup\int_X|f_n-f|d\mu$

As $g\in\mathcal{L}^1$, $\displaystyle\limsup_{n\to\infty}\int_X|f_n-f|d\mu\leq$ implying that $\displaystyle\limsup\int_X|f_n-f|d\mu$

Theorem

1. Suppose $f:X\to[0,\infty]$ is measurable, $E\in\sigma$ with μ(E) > 0 such that $\displaystyle\int_Efd\mu=0$. Then f(x) = 0 almost everywhere E
2. Let $f\in\mathcal{L}^1(X)$ and let $\displaystyle\int_Efd\mu=0$ for every $E\in\sigma$. Then, f = 0 almost everywhere on X
3. Let $f\in\mathcal{L}^1(X)$ and $\displaystyle\left|\int_Xfd\mu\right|=\int_X|f|d\mu$ then there exists constant α such that | f | = αf almost everywhere on E

Proof

1. For each $n\in\mathbb{N}$ define $A_n=\{x\in E|f(x)>\frac{1}{n}\}$. Observe that $A_n\uparrow E$
   but $\frac{1}{n}\mu(A_n)\leq\displaystyle\int_{A_n}fd\mu\leq\int_Efd\mu=0$ Thus μ(A_n) = 0 for all n, by continuity, f = 0 almost everywhere on E
2. Write $\displaystyle\int_Efd\mu=\int_Eu^+d\mu-\int_Eu^-d\mu+i\left(\int_Ev^+d\mu-\int_Ev^-d\mu\right)$, where u ⁺ ,u ⁻ ,v ⁺ ,v ⁻ are non-negative real measurable.
   Further as $\displaystyle\int_Eu^+d\mu,\int_E(-u^-)d\mu$ are both non-negative, each of them is zero. Thus, by applying part I, we have that u ⁺ ,u ⁻ vanish almost everywhere on E. We can similarly show that v ⁺ ,v ⁻ vanish almost everywhere on E.