Let $(X,𝔛,\mu )$ be a standard probability space. We say that a sub-σ-algebra $𝔅$ of $𝔛$decomposes μ in an ergodic way if any regular conditional probability ${}^{𝔅}P$ with respect to $𝔅$ andμ satisfies, for μ-almost every x∈X, $\forall B\in 𝔅,{}^{𝔅}P(x,B)\in \{0,1\}$. In this case the equality $\mu (·)={\int }_X{}^{𝔅}P(x,·)\mu \left(\mathrm{d}x\right)$, gives us an integral decomposition in “$𝔅$-ergodic” components. For any sub-σ-algebra $𝔅$ of $𝔛$, we denote by $\overline{𝔅}$ the smallest sub-σ-algebra of $𝔛$ containing $𝔅$ and the collection of all setsAin $𝔛$ satisfyingμ(A)=0. We say that $𝔅$ isμ-complete if $𝔅=\overline{𝔅}$. Let $\{{𝔅}_{i}i\in I\}$ be a non-empty family...

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/(d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right))+...$, where $d_j\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}\left(x\right)\ge d_j\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty }{d}_n^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_H$ to denote the Hausdorff...

We show that a σ-algebra 𝔹 carries a strictly positive continuous submeasure if and only if 𝔹 is weakly distributive and it satisfies the σ-finite chain condition of Horn and Tarski.

A topological space Y is said to have (AEEP) if the following condition is satisfied: Whenever (X,) is a measurable space and f,g: X → Y are two measurable functions, then the set Δ(f,g) = x ∈ X: f(x) = g(x) is a member of . It is shown that a metrizable space Y has (AEEP) iff the cardinality of Y is not greater than $2^{\aleph ₀}$.

In this paper, we define a $$-integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $\mu$ is compatible with its topology in the sense that every open set is $\mu$-measurable. We prove that the $$-integral is equivalent to $\mu$-integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.