Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $p\in (0,1)$. For each $k\in\mathbb{N}$ and tuple $(\varepsilon_1,\ldots,\varepsilon_k)\in\{0,1\}^k$ denote $$ S_{\varepsilon_1,\ldots,\varepsilon_k}=\left\{\sum\limits_{j=1}^\infty x_j 2^{-j}: x\in\{0,1\}^\mathbb{N}\;\wedge\;x_1=\varepsilon_1,\ldots,x_k=\varepsilon_k\right\} $$ $$ \mathcal{S}=\{S_{\varepsilon_1,\ldots,\varepsilon_k}:(\varepsilon_1,\ldots,\varepsilon_k)\in\{0,1\}^k,\;k\in\mathbb{N}\}\cup\{\varnothing\} $$ One can show that $\mathcal{S}$ is a semiring of diadic segments. By $\mathfrak{B}(\mathcal{S})$ we denote minimall $\sigma$-algebra that contains $\mathcal{S}$. Define measure $m_p:\mathcal{S}\to\mathbb{R}_+$ by equalities $$ m_p(S_{\varepsilon_1,\ldots,\varepsilon_k})=\prod\limits_{i=1}^k p^{\varepsilon_i}(1-p)^{1-\varepsilon_i}\\ m_p(\varnothing)=0 $$ Consider Lebesgue extension of $\lambda_p$ of measure $m_p$. It is defined on some $\sigma$-algebra $\mathfrak{M}_p(\mathcal{S})$. Since I want to consider this extensions for different $p$ I will restrict $\lambda_p$ to $\mathfrak{B}(\mathcal{S})$ and denote the resulting measure $\mu_p$.

I will be happy to get answers to some of the following questions.

1) How to prove that $m_p$ is $\sigma$-additive?

2) Is it true that $\mu_p$ is regular?

3) Is it true that $\mathfrak{M}_{p_1}(\mathcal{S})\neq \mathfrak{M}_{p_2}(\mathcal{S})$ for $p_1\neq p_2$?

4) Does there exist explicit construction of the set $B\in\mathfrak{B}(\mathcal{S})$ such that $\mu_{p_1}(B)=1$ and $\mu_{p_2}(B)=0$ for $p_1\neq p_2$.

Less concrete questions are:

1) Does this type of measures have a name?

2) Where can I find articles about their properties?

Thank you for taking time.

share|improve this question
1  
You are making the digits in the binary expansion i.i.d. bernoulli(p) ? –  mike Dec 19 '12 at 12:30
    
@maike yes, you are right –  Norbert Dec 19 '12 at 15:10
add comment

1 Answer

up vote 1 down vote accepted
+250

You want "credible and/or official sources"??? You mean on-line sources are not wanted? There is this....

(22.35) through (22.44) in Real and Abstract Analysis by Hewitt & Stromberg (Springer, 1965)

share|improve this answer
    
on-line sources are also good –  Norbert Dec 21 '12 at 22:53
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.