# Properties of certain type of measures

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.

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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 –  userNaN Dec 19 '12 at 15:10

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)

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on-line sources are also good –  userNaN Dec 21 '12 at 22:53