Measure Theory

Consider a discrete-time random walk on the real numbers ℝ. Let: s ∈ ℝ be the initial position of the walker. L < s < U be the lower and upper absorbing barriers, respectively. p ∈ (0, 1) be th...

Suppose we partition the reals into two sets $A$ and $B$ that are dense (with positive Lebesgue measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$, where Lebesgue measure $\lambda$ restr...

Suppose we partition the reals into two sets $A$ and $B$ that are dense and has positive Lebesgue measure in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$, where Lebesgue measure $\lambda$ rest...

My simple question might suggest I'm missing something quite basic. I, therefore, apologize in advance.In Prop 1.15, Folland defines a pre-measure on h-intervals (left open right closed intervals).1.1...

a) Use the sequence of functions $(f_{n})$ to establish that the inequality in Fatou's lemma can be strict.b) Use the sequence of functions $(f_{n})$ to establish that the motto of Fatou is false in g...

1) Let $(X,\Sigma,\mu)$  be a measure space and for each n suppose that $f_{n}:X\rightarrow \mathbb{R} $ is an integrable function such that $\sum_{n=1}^{\infty } \int |f_{n}|du$ is convergent.Prove t...

Let X be a non-empty set and let E denote the collectionof all unitary subsets of X, that is,E = {{x} ; x $\in$ X}.(a) What is the $\sigma $ -algebra generated by E?(b) If A is the $\sigma $-algebra g...

Let S={$\varnothing $ ,(-$\infty $ ,0), ([0,+$\infty $ ), $\mathbb{R} $ }(wich is a $\sigma $ -algebra on $\mathbb{R}$), then a function f:$\mathbb{R} $ -> $\mathbb{R}$ is S- measurable if and only...

Let $X$ a set and $Y$ a subset of $X$. Let $B$ a sigma algebra over $X$ and $F=\left\{ E\cap Y:\,E\in B\right\}$ a sigma algebra over $Y$.If $B=\sigma(\mathcal{E}_{1})$ and $\mathcal{E}_{2}=\left\{ E\...

Calculate the integral given in the picture where Lambda^2 denotes the Lebesgue measure on R^2.The method is important to me as I need to solve multiple of these types of questions.

Let $(\Omega, \mathcal{F})$ be a measurable space and $\mathcal{E} \subset \mathcal{P}(\Omega)$ a generator of $\mathcal{F}$, i.e. $\mathcal{F}$ is the intersection of all $\sigma$-algebras that conta...

If $(\Omega, \mathcal{A}, P)$ is a probability space without atoms, then for every $a \in[0,1]$ there exists at least one set $A \in \mathcal{A}$ of probability $P(A)=a$. The question comes with a hin...

See the attachment below.If you're unable to solve it try to find a research paper.

Motivation:Some sets have a Hausdorff Dimension $\alpha$ but have a zero-dimensional Hausdorff Measure. These sets may have another dimension function https://en.m.wikipedia.org/wiki/Dimension_functio...

Question: How do we describe an intuitive arithmetic mean that gives a defined value between the infimum and supremum of a function's range (and coincides with other arithmetic means) without peice-wi...

Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$, or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such...

Suppose that $\mu,\nu, \lambda$ are $\sigma$-finite measures on a measurable space $(X,\mathcal{M})$ and that Prove that  

Suppose that $\mu$ and $\nu$ are finite measures on a measurable space $(X,\mathcal{M})$. Prove that either $\nu \perp \mu$ or there exists $\epsilon > 0$ and a measurable set $E \subseteq X$ such...