Borel Sigma-Algebra on Euclidean Space by Monotone Class

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Theorem

Let $\left({\R^n, \tau}\right)$ be the $n$-dimensional Euclidean space.


Then:

$\mathcal B \left({\R^n, \tau}\right) = \mathfrak m \left({\tau}\right)$

where $\mathcal B$ denotes Borel $\sigma$-algebra, and $\mathfrak m$ denotes generated monotone class.


Proof

Let $U \in \tau$ be an open set, and define $C$ by:

$C := X \setminus U$

hence $C$ is a closed set.


Further, define, for all $n \in \N$:

$C_n := \displaystyle \bigcup_{c \mathop \in C} B \left({c; \frac 1 n}\right)$

where $B$ denotes open ball.

The $C_n$ are open sets, being the union of open balls.


It is clear that $C \subseteq C_n$ for all $n \in \N$.

Conversely, as $U$ is open, for any $u \in U$ (that is, $u \notin C$), find $n \in \N$ such that:

$B \left({u; \dfrac 1 n}\right) \subseteq U$

as is possible from the definition of open set in a metric space.

Thus, for all $c \in C = X \setminus U$, this means:

$d \left({u, c}\right) \ge \dfrac 1 n$

whence $u \notin C_n$.


That is, we have established that:

$c \in C \iff \forall n \in \N: c \in C_n$

Phrased in terms of intersection, this means:

$C = \displaystyle \bigcap_{n \mathop \in \N} C_n$


Thus, since $C_n \in \tau \subseteq \mathfrak m \left({\tau}\right)$:

$C \in \mathfrak m \left({\tau}\right)$

Now define:

$\complement_X \left({\tau}\right) := \left\{{X \setminus U: U \in \tau}\right\}$


Then we have shown:

$\mathfrak m \left({\tau \cup \complement_X \left({\tau}\right)}\right) \subseteq \mathfrak m \left({\tau}\right)$

and the reverse inclusion (and hence equality) follows from Generated Monotone Class Preserves Subset.


Now applying Generated Sigma-Algebra by Generated Monotone Class: Corollary:

$\sigma \left({\tau}\right) = \mathfrak m \left({\tau \cup \complement_X \left({\tau}\right)}\right)$

Combining these two equalities gives the result, by definition of Borel $\sigma$-algebra.

$\blacksquare$


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