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### Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
18k views

### Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue ...
3k views

### Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel $\sigma$-...
839 views

### Lebesgue measure of algebraic irrational numbers in $\mathbb{R}$

Find all Lebesgue measurable subsets $A \subset\mathbb{R}$ such that all $B\subset A$ is measurable. I argued that if the measure is positive then $A$ is an interval so we can construct the Vitali ...
646 views

### Any positive measure subset of $\mathbb R$ contains a positive measure Cantor set

A question asks to show any positive measure subset of $\mathbb R$ contains a positive measure Cantor set. How to start with this? I have been staring on this for a while, but can not come up with any ...
384 views

### Let $A\subseteq\Bbb R$ with $\lambda^*(A)>0$. Show that there exists a nonmeasurable $B\subseteq\Bbb R$ s.t. $B\subseteq A$

Let $A$ be a subset of $\Bbb R$ with $\lambda^*(A)>0$. Show that there exists a nonmeasurable subset $B$ of $\Bbb R$ s.t. $B$ is a subset of $A$ I'm a little confused where to start with this ...
106 views

### Prove or disprove $\nu(E)=\lambda(f(E))$ is a measure provided that $f$ is nondecreasing and satisfies the N-condition.

Suppose $f$ is a non-decreasing continuous function from $[a,b]$ to $\mathbb{R}$, and $\lambda$ is the Lebesgue measure in $\mathbb{R^1}$. Also, $f$ satisfies the property that $f$ maps Lebesgue ...
111 views

### A question about Lebesgue outer measure

Is there a set $X \subseteq \mathbb{R}$ such that Lebesgue outer measure is countably additive on subsets of $X$? Of course we also require $X$ to have positive outer measure.
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### Measure $\mu$ defined through homeomorphism $f$, $\mu(A)=m_n(f(A))$

I was asked to verify whether the next claim is true or false: Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a homeomorphism and define $\mu(A)=m_n(f(A))$, where $m_n$ is the $n$-dimensional ...
39 views

### Borel set's incompleteness

I was told that while M, class of all Lebesgue measurable set is complete, sigma field generated by all open set B is not. Can someone give me an example of interval which is inside M yet is not ...