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2 — Algebras, Measures, Length Measure, Continuity, σ-Algebras

Lecture 2 deepens the algebra/measure part from Lec 1—formalizes algebras on $\mathbb{R}$ via finite unions of semiopen intervals, defines length pre-measure (b-a), proves countable additivity on those sets, and introduces (\sigma)-algebras and extension to Borel sets.

We work on a general measurable space $(\Omega, \mathcal{A}, \mu)$ where
$\mathcal{A}$ is a collection of subsets of $\Omega$ and $\mu$ is a measure.

1. Algebras of Sets

A collection $\mathcal{A} \subseteq 2^\Omega$ is an algebra if:

  1. Contains whole space:
    $\Omega \in \mathcal{A}$.

  2. Closed under complement:
    If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$.

  3. Closed under finite intersections:
    If $A,B \in \mathcal{A}$, then $A \cap B \in \mathcal{A}$.

From these, closure under finite unions follows via De Morgan: \(A \cup B = (A^c \cap B^c)^c.\)

2. Main Example of an Algebra on $\mathbb{R}$

Let \(\mathcal{A} = \left\{ \bigcup_{i=1}^n (a_i, b_i] : a_i < b_i,\ \text{intervals disjoint} \right\}.\)

This is an algebra consisting of finite unions of semiopen intervals.

A graphical example shown in class (page 1 of PDF) illustrated how non-disjoint intervals can always be merged into disjoint ones.

3. Measures on an Algebra

A function $\mu : \mathcal{A} \to [0,\infty]$ is a measure if:

  1. Non-negativity: $\mu(A) \ge 0$.
  2. Null empty set: $\mu(\varnothing) = 0$.
  3. Countable additivity on disjoint sets:
    If $A_i \in \mathcal{A}$ are disjoint and $\bigcup_{i=1}^\infty A_i \in \mathcal{A}$, then
    \(\mu\Big( \bigcup_{i=1}^\infty A_i \Big) = \sum_{i=1}^\infty \mu(A_i).\)

Note: On an algebra (not yet a σ-algebra), $\bigcup_{i=1}^\infty A_i$ may not belong to $\mathcal{A}$, so (3) only applies when the union stays inside $\mathcal{A}$.

4. Length Measure (Lebesgue Pre-Measure on Semiopen Intervals)

Define for a semiopen interval: \(\mu((a,b]) = b - a.\)

For a finite disjoint union: \(\mu\Big( \bigcup_{i=1}^n (a_i, b_i] \Big) = \sum_{i=1}^n (b_i - a_i).\)

This is the Lebesgue pre-measure on $(a,b]$-intervals.

5. General Form of Measures of the Type $M([a,b]) = F(b) - F(a)$

Let $F : \mathbb{R} \to \mathbb{R}$ be:

  1. Non-decreasing.
  2. Right-continuous.

Then define: \(M((a,b]) = F(b) - F(a).\)

This produces a measure on the algebra generated by semiopen intervals.

This generalizes Lebesgue measure (for which $F(x)=x$).

6. Verifying Countable Additivity for Length Measure

Assume: \(A = \{ \bigcup_{i=1}^n (a_i,b_i] \subset (0,1]\}, \quad \mu((0,1]) = 1.\)

Suppose $A_i \in \mathcal{A}$ are pairwise disjoint and
$\bigcup_{i=1}^\infty A_i \in \mathcal{A}$.

Then

\[\mu( \bigcup_{i=1}^\infty A_i ) = \mu(\bigcup_{i=1}^n A_i ) + \mu(\bigcup_{i=n+1}^\infty A_i).\]

Let \(B_n = \bigcup_{i=n+1}^\infty A_i.\)

Then

  • $B_{n+1} \subseteq B_n$ (decreasing sequence).
  • $\bigcap_{n=1}^\infty B_n = \varnothing$.
  • $\mu(B_n) \downarrow 0$.

Hence

\[\sum_{i=1}^\infty \mu(A_i) = \lim_{n\to\infty} \sum_{i=1}^n \mu(A_i) = \mu\Big( \bigcup_{i=1}^\infty A_i \Big).\]

7. Example of Decreasing Sets with Vanishing Measure

Let

\[D_n = \left( 0, \frac{1}{n} \right],\]

a decreasing sequence ($D_{n+1} \subset D_n$) with: \(\mu(D_n) = \frac{1}{n} \downarrow 0.\)

Thus, measure is continuous from above for decreasing sequences whose intersection is empty.

8. What If $\mu(B_n)$ Does Not Go to 0?

Suppose instead that: \(\mu(B_n) \not\downarrow 0.\)

Then ∃ $a > 0$ such that: \(\mu(B_n) \ge a \quad \forall n.\)

If each $B_n$ is compact (closed and bounded in $\mathbb{R}$), and
\(B_{n+1} \subseteq B_n,\) then the nested intersection property for compact sets yields: \(\bigcap_{n=1}^\infty B_n \ne \varnothing.\)

So the only way for compact nested sets to have intersection empty is that their measures must shrink to 0.

This illustrates:

Continuity from above fails only when the sets are not compact (or not “small enough”).

The handwritten notes on pages 2–3 show explicit constructions of $C_n \subset B_n$ and their closures to force compactness and derive contradiction.

9. σ-Algebra $\mathcal{A}^*$

A collection $\mathcal{A}^*\subseteq 2^\Omega$ is a σ-algebra if:

  1. $\Omega \in \mathcal{A}^*$.
  2. $A \in \mathcal{A}^* \implies A^c \in \mathcal{A}^*$.
  3. $A_i \in \mathcal{A}^* \implies \bigcup_{i=1}^\infty A_i \in \mathcal{A}^*$.

10. Measures on σ-Algebras

A measure $\mu$ on a σ-algebra $\mathcal{A}^*$ satisfies:

  1. $\mu(A)\ge 0$.
  2. $\mu(\varnothing)=0$.
  3. Countable additivity for any countable disjoint family: \(\mu\Big( \bigcup_{i=1}^\infty A_i \Big) = \sum_{i=1}^\infty \mu(A_i).\)

11. From an Algebra to a σ-Algebra

Given an algebra $\mathcal{A}$, the smallest σ-algebra containing it is:

\[\sigma(\mathcal{A}) = \text{the minimal σ-algebra with } \mathcal{A}\subseteq \sigma(\mathcal{A}).\]

In our interval example over $\mathbb{R}$,
\(\sigma(\mathcal{A}) = \text{Borel σ-algebra } \mathcal{B}(\mathbb{R}).\)

This is called the Borel σ-algebra, generated by open intervals or semiopen intervals.

Carathéodory’s extension theorem allows one to extend a pre-measure on $\mathcal{A}$ (e.g., length) to a full measure on $\sigma(\mathcal{A})$.

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