We want a measure that has the following properties

1. Respects lengths of intervals meaning that $\mu (p[a,b])=b-a,\mathrm{\forall }0\le a\le b\le 1$
2. it should satisfy countable additivity i.e

3. It should be translation invariant meaning that if $A\subseteq [0,1]$ and we define $A_x=A+x$ to be A shifted rightward b x then $m(A)=\mu (A_x)$

There is no function $\mu :\mathcal{P}([0,1])\to [0,1]$ that satisfies all three of the properties we want in a measure

a $\sigma$-algebra of sets is a collection of subsets in ${\mathbb{R}}^d$ that is closed under countable unions, intersections and complements

A measure space is a triple $(\mathrm{\Omega },\mathcal{F},\mu )$ satisfying the following axioms:

• $\mathrm{\Omega }$, the state space or outcome space, is a nonempty set.
• $\mathcal{F}$, a set of measurable subsets or events, is a σ-field or σ-algebra over $\mathrm{\Omega }$ (those terms will be used interchangeably). In other words, $\mathcal{F}$ is a nonempty collection of subsets of $\mathrm{\Omega }$ which is closed under complementation and countable union, so if $A\in \mathcal{F}$, then $A^c=\mathrm{\Omega }\setminus A$ is also in $\mathcal{F}$, and if $A_i\in \mathcal{F}$, then $\bigcup _{i=1}^{\mathrm{\infty }}{A}_i$ is also in $\mathcal{F}$.
• $\mu$, the measure, is a map $\mathcal{F}\to [0,\mathrm{\infty }]$ which is not infinite everywhere and is countably additive. In other words, if $A_i\in \mathcal{F}$ are disjoint, then $\mu \left(\underset{i=1}{\overset{\mathrm{\infty }}{⨆}}{A}_i\right)=\sum _{i=1}^{\mathrm{\infty }}\mu (A_i)$.

A measure $\mu$ is a probability measure (which we will denote $\mathbb{P}$) if $\mu (\mathrm{\Omega })=1$.

Given a measure space $(\mathrm{\Omega },\mathcal{F},\mu )$, the following must hold:

1. $\mathrm{\varnothing }$ and $\mathrm{\Omega }$ are in $\mathcal{F}$,
2. $\mu (\mathrm{\varnothing })=0$,
3. (continuity from below) if $A_i\uparrow A$ – that is, $A_1\subseteq A_2\subseteq \cdots$ and $A=\bigcup _{i=1}^{\mathrm{\infty }}{A}_i$ – then $\mu (A_i)\uparrow \mu (A)$,
4. (continuity from above) if $A_i\downarrow A$ – that is, $A_1\supseteq A_2\supseteq \cdots$ and $A=\bigcap _{i=1}^{\mathrm{\infty }}{A}_i$ – and also $\mu (A_1)<\mathrm{\infty }$, then
   $\mu (A_i)\downarrow \mu (A)$,
5. for any $\mathrm{\Omega }$, $\mathcal{F}=\{\mathrm{\varnothing },\mathrm{\Omega }\}$ is a valid $\sigma$-field, and so is $\mathcal{P}(\mathrm{\Omega })$.

we note that $\mu (A_i)\downarrow \mu (A)$ or $\mu (A_i)\uparrow \mu (A)$ is basically saying $\underset{i\to \mathrm{\infty }}{lim}\mu (A_i)=\mu (A)$.

The Borel $\sigma$-field ${\mathcal{B}}_{\mathbb{R}}$ is the smallest $\sigma$-field over $\mathbb{R}$ that contains $\mathcal{I}$, the set of all open intervals on $\mathbb{R}$.

A Stieltjes measure function on $\mathbb{R}$ is a function $F:\mathbb{R}\to \mathbb{R}$ which is non decreasing and right continuous

(Lebesgue-Stieltjes):
For any Stieltjes measure function F, there is a unique measure $\mu ={\mu }_F$ on $(\mathbb{R},{\mathcal{B}}_{\mathbb{R}})$ satisfying $\mu ((a,b])=F(b)-F(a)$ for all $a,b\in \mathbb{R}$.

Our measure $\mu ={\mu }_F$ has the following properties on $A$:

1. $\mu$ is monotone and finitely additive over $A$, meaning that for any disjoint $A_1,\dots ,A_n\in A$, $\mu (\underset{i}{⨆}{A}_i)=\sum _{i=1}^n\mu (A_i)$.
2. $\mu$ is finitely subadditive over $A$, meaning that for any $A_1,\dots ,A_n\in A$, $\mu (\underset{i}{⨆}{A}_i)\le \sum _{i=1}^n\mu (A_i)$.
3. $\mu$ is countably subadditive over $\mathcal{I}$: if $A,A_i\in \mathcal{I}$ (where $i$ ranges over the positive integers) and $A\subseteq \bigcup _i{A}_i$, then $\mu (A)\le \sum _i\mu (A_i)$.
4. $\mu$ is countably additive on $A$.

A $\pi$-system P on a set $\mathrm{\Omega }$ is a nonempty collection of subsets of $\mathrm{\Omega }$, such that for any $A,B\in \mathcal{P}$, $A\cap B\in \mathcal{P}$.

• A $\lambda$-system $\mathcal{L}$ on a set $\mathrm{\Omega }$ is a collection of subsets of $\mathrm{\Omega }$ satisfying the following conditions:
• $\mathrm{\Omega }\in \mathcal{L}$.
• For any $A,B\in \mathcal{L}$ where $B\subseteq A$, $A\setminus B\in \mathcal{L}$.
• If we have a sequence $A_n\in \mathcal{L}$, and $A_n\uparrow A$, then $A\in \mathcal{L}$.

If a $\lambda$-system $\mathcal{L}$ is also a $\pi$-system, then $\mathcal{L}$ is a $\sigma$-algebra.

(Dynkin's $\pi -\lambda$ theorem)
If P is a $\pi$-system, L is a $\lambda$-system, and $P\subseteq L$, then $\sigma (P)\subseteq L$.