Set of borel measurable functions

saiaN · **Joined:** Fri, 18 May 2012 03:00:09 UTC **Posts:** 3

Consider collections $\mathcal{V}$ of functions from $\mathbb{R}$ to $\mathbb{R}$ satisfying the following conditions:

(a) $\mathcal{V}$ is a vector space

(b) $\mathcal{V}$ contains the continuous functions

(c) If $(f_{n})_{n}$ is an increasing sequence of nonnegative functions in $\mathcal{V}$ and if $\lim_{n \rightarrow \infty} f_{n}(t)$ exists and is finite for all $t \in \mathbb{R}$ , then $\lim_{n\rightarrow \infty} f_{n}(t) \in \mathcal{V}$ .

Show that the collection $\mathcal{V}_{0}$ consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define $\mathcal{A}= \{ A \subseteq \mathbb{R}: \chi_{a} \in \mathcal{V} \}$ . Show that $\mathcal{A}$ contains the interval $(-\infty,a)$ , and then contains the Borel sets).

$\chi_{A} :$ characteristic function of

.

$\chi_{A}(x)=1$ if $x \in A$ , $\chi_{A}(x)=0$ if $x \notin A,$

I show that $(-\infty,a)$ is in $\mathcal{A}$ , but i try prove $\mathcal{A}$ is a sigma-algebra but the union countable give me troubles, some help for it?

any help is apreciated, my first post

outermeasure · **Posted:** Fri, 18 May 2012 05:07:26 UTC

saiaN wrote:

Consider collections $\mathcal{V}$ of functions from $\mathbb{R}$ to $\mathbb{R}$ satisfying the following conditions:

(a) $\mathcal{V}$ is a vector space

(b) $\mathcal{V}$ contains the continuous functions

(c) If $(f_{n})_{n}$ is an increasing sequence of nonnegative functions in $\mathcal{V}$ and if $\lim_{n \rightarrow \infty} f_{n}(t)$ exists and is finite for all $t \in \mathbb{R}$ , then $\lim_{n\rightarrow \infty} f_{n}(t) \in \mathcal{V}$ .

Show that the collection $\mathcal{V}_{0}$ consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define $\mathcal{A}= \{ A \subseteq \mathbb{R}: \chi_{a} \in \mathcal{V} \}$ . Show that $\mathcal{A}$ contains the interval $(-\infty,a)$ , and then contains the Borel sets).

$\chi_{A} :$ characteristic function of

.

$\chi_{A}(x)=1$ if $x \in A$ , $\chi_{A}(x)=0$ if $x \notin A,$

I show that $(-\infty,a)$ is in $\mathcal{A}$ , but i try prove $\mathcal{A}$ is a sigma-algebra but the union countable give me troubles, some help for it?

any help is apreciated, my first post

$\displaystyle\chi_{\bigcup A_n}=\mathop{\uparrow\text{-}\lim}_n \chi_{\bigcup_{j=1}^n A_j}$ .

saiaN · **Joined:** Fri, 18 May 2012 03:00:09 UTC **Posts:** 3

outermeasure wrote:

$\displaystyle\chi_{\bigcup A_n}=\mathop{\uparrow\text{-}\lim}_n \chi_{\bigcup_{j=1}^n A_j}$ .

sorry but what is the arrow?, you could be more explicit please
thx

Shadow · **Posted:** Sat, 19 May 2012 00:09:20 UTC

saiaN wrote:

outermeasure wrote:

$\displaystyle\chi_{\bigcup A_n}=\mathop{\uparrow\text{-}\lim}_n \chi_{\bigcup_{j=1}^n A_j}$ .

sorry but what is the arrow?, you could be more explicit please
thx

monotone increasing.

saiaN · **Joined:** Fri, 18 May 2012 03:00:09 UTC **Posts:** 3

great, just keep prove that $\chi_{\cup_{j=1}^{n}A_{j}}$ is in $\mathcal{V}$
I will try to prove

outermeasure · **Posted:** Sun, 20 May 2012 08:51:29 UTC

saiaN wrote:

great, just keep prove that $\chi_{\cup_{j=1}^{n}A_{j}}$ is in $\mathcal{V}$
I will try to prove

Have you seen the Dynkin's $\pi\text{-}\lambda$ theorem (or $\pi\text{-}d$ lemma)?

Note that the exercise is a special case of the following version of the monotone class theorem:

Theorem Let

be a collection of bounded functions $\Omega\to\mathbb{R}$ closed under pointwise product, and let $\mathcal{F}$ be the $\sigma$ -algebra generated by

. Let $V\supseteq S$ be a vector space (over $\mathbb{R}$ ) of bounded real-valued functions on $\Omega$ such that
(a)

contains the constant functions; and
(b) if $(f_n)\subseteq V$ with $\displaystyle\sup_n\sup_\omega\lvert f_n(\omega)\rvert<+\infty$ and $0\leq f_1\leq f_2\leq f_3\leq\dots$ , then $\displaystyle f:=\mathop{\uparrow\text{-lim}}_n f_n\in V$ .
Then

contains every bounded $\mathcal{F}$ -measurable real-valued functions on $\Omega$ .

The proof is essentially the same.

S.O.S. Mathematics CyberBoard

Set of borel measurable functions

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