#import "/book.typ": book-page
#import "../../../templates/conf.typ": *
#import "@preview/mitex:0.2.4": *
#show: thmrules.with(qed-symbol: $square$)

#show: book-page.with(title: "1-D Lebesgue Measure")

= 1-D Lebesgue Measure

Let $F: RR -> RR$ be increasing and right-continuous.
Consider intervals of the type "h-intervals":
sets of the form $]a, b]$ or $[a, +infinity[$ or $diameter$ for $-infinity <= a < b <= +infinity$.

#prp[

$cal(A) = {"finite disjointegral unions of h-intervals"}$ is an algebra, and
$
cases(
mu_0(diameter) = 0,
mu_0(union.sq.big_1^infinity (a_j, b_j]) :=  sum_1^infinity F(b_j) - F(a_j)
)
$
defines a premeasure on $cal(A)$ .

]

Since $sigma(cal(A)) = cal(B)_(RR^1)$ , we can apply _Hahn extension_ to
extend $mu_0$ to a unique Borel measure on $RR$ , called the Lebesgue-Stieltjes measure $mu_F$.

#prp[

$mu_F = mu_G <=> F = G + "constant"$.

]

We write $mu_F equiv mu$ and
let $cal(M)_mu := {mu^* "-measurable subsets"}$ which is the domain of $macron(mu)$ .
Then $forall E in cal(M)_mu$ ,
$
mu(E) = inf{sum_1^infinity mu(]a_j, b_j]) : E subset union.big_1^infinity (a_j, b_j]}.
$

#lem[

$
mu(E) = inf{sum_1^infinity mu((a_j, b_j)) : E subset union.big_1^infinity (a_j, b_j)}
$

]

#prf[

Suppose $E subset union.big_1^infinity (a_j, b_j)$.
Since $(a_j, b_j) = union.big_1^infinity (a_j, b_j - 1/n] =: union.sq.big_(j, n = 1)^infinity I_(j, n)$ for some disjoint h-intervals $I_(j, n)$, so
$
mu(E) <= mu(union.sq.big_(j, n = 1)^infinity I_(j, n)) = mu(union.big_1^infinity (a_j, b_j)).
$

On the other hand, by the definition of $mu(E)$ , $forall epsilon > 0, exists {(a_j, b_j]}_1^infinity$ , s.t.
$
mu(E) >= sum_1^infinity mu((a_j, b_j]) - epsilon,
$
where $mu((a_j, b_j]) = F(b_j) - F(a_j)$.

For $F$ right-countinuous, so $exists delta_j >0$, s.t.
$
F(b_j + delta_j) - F(b_j) < epsilon/2^j,\
=> mu((b_j, b_j + delta_j)) < epsilon/2^j .
$
Then $mu((a_j, b_j + delta_j)) <= mu((a_j, b_j]) + epsilon/2^j$ , so
$
mu(E) >= sum_1^infinity mu((a_j, b_j]) - epsilon >= sum_1^infinity mu((a_j, b_j + delta_j)) - 2epsilon
$
where $E subset union.big_1^infinity (a_j, b_j + delta_j)$ . Sending $epsilon -> 0$ , the result follows.

]

Let $mu_F :=$ the (implies uniqueness) measure on $sigma({"h-integralerbals"})$ obtained via Hahn extension.

Convention: Denote by $mu_F$ its own completion.

Recall:
$
mu(E) = inf{sum_(j=1)^(infinity) mu((a_j, b_j)): E subset union.sq.big_(j=1)^(infinity) (a_j, b_j), forall E in cal(M)_mu},
$
and the fact that
$
U subset RR "is open" <=> U = union.sq.big_(j=1)^(infinity) I_j,
$
where ${I_j}$ are disjoint open intervals. We may deduce that $mu := mu_F$ , the Lebesgue-Stieltjes measure, is regular.

#thm[

Let $E in cal(M)_mu$ be arbitrary, then
$
mu(E) &= inf{mu(O): O "is open", O supset E} quad & "(outer regular)" \
      &= sup{mu(K): K "is compact", K subset E}quad & "(inner regular)"
$

]

Proof is left as an exercise.

For sets $A, B$ , $A Delta B := (A \\ B) union (B \\ A)$ .
In fact, Borel set ($E in cal(B)_RR$) $Delta$ "good set" = null set. 

#def[

A $G_delta$ - set is a countable insections of open sets. An $F_sigma$ - set is a countable unions of closed sets.

]

"Every Borel set on $RR$ is nearly a (finite unions of) open intervals." ------ _Littlewood's First Principle_

#thm[

Let $E subset RR$ , TFAT(the following are equivalent)

1. $E in cal(M)_mu$,
2. $E = V \\ N_1$ with $V$ is $G_delta$-set, $mu(N_1) = 0$,
3. $E = H union N_2$ with $H$ is $F_sigma$-set, $mu(N_2) = 0$.

]

#prf[

2, 3 $=>$ 1:

Since $mu$ is complete on $cal(M)_mu$ and $G_delta, F_sigma$-sets are Borel sets. Hence $E in cal(M)_mu$ .

1 $=>$ 2 & 3:

By regularing of $mu$, $forall E in cal(M)_mu, forall j in NN, exists$ open set $O_j subset RR$ , compact set $K_j subset RR$ , s.t.
$
cases(
  K_j subset E subset O_j  \
  mu(O_j) - 2^(-j) <= mu(E) <= mu(K_j) + 2^(-j) quad "if" mu(E) < infinity .
)
$
Then take $V = sect.big_(j=1)^(infinity) O_j, H = union.big_(j=1)^(infinity) K_j$ .

TODO: the general case: $mu(E) = infinity$ .

]

#def[

When $F$ is identity mapping, $mu_F = cal(L)^1$ is the Lebesgue measure on $RR$. Its domain $cal(M)_mu$ is the set of Lebesgue measurable sets.

]

In fact, we have $cal(M)_mu supset cal(B)_RR$ and the supset sign is strict.

About Lebesgue measurable, we have some key (pathological) examples.

#exm[

There are open, dense subsets of $(0, 1)$ of arbitrary small $cal(L)^1$-measure. In orther words, topological big $arrow.r.double.not$ measure-theoretically big.

]

A specific example is as follows.

#exm[

Take $q_1, q_2, ...$ be an enumenation of $QQ sect (0, 1)$ . Fix any $epsilon$ , and let
$
O = union.big_(j=1)^(infinity) (q_j - epsilon/(2j), q_j + epsilon/(2j))
$
then
$
cal(L)^1(O) <= sum_(j=1)^(infinity) mu((q_j - epsilon/(2j), q_j + epsilon/(2j))) <= 2 epsilon.
$


]