# Tagged Questions

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### Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
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### Square of absolute value of a function different than square of function

How come if f is measurable, we might have $|f|^2\neq f^2$? Can you provide an example? I think it is true if f is real.
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### Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?

Is there a nowhere differentiable but continuous everywhere function which is monotone in some small interval however small it is? Until now I have seen only the Weierstrass function and it seems to ...
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### A wonderful property of real line!

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
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### Finding an example where a measure is not unique

Let $(X, \mathcal{M})$ be a measurable space. Let $\mu$, $\nu$ be measures defined on $\mathcal{M}$. (a) For $A \in \mathcal{M}$ define $\lambda(A)=\mu(A)+ \nu(A)$. Prove that $\lambda$ is a ...
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### Does $f<\infty$ a.s. imply that $f$ is integrable?

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f\colon\Omega\to\overline{\mathbb{R}}$ measurable. Does then $f<\infty$ a.s. imply that $f$ is integrable? I think no, but ...
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### Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
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### Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.

Show that there exists a not decreasing function $f:(a,b)\rightarrow\mathbb{R}$ continuous on $(a,b)\setminus D$ and discontinuous on $D$ where $D$ is a countably infinite subset of $(a,b)$. This is ...
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### An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
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### Measure, absolutely continuous on boundary

Let $\mu$ be a finite nonnegative Borel measure on $\mathbb R^2_+=[0,+\infty) \times [0,+\infty)$ such that $\mu( \partial \mathbb R^2_+)=0$, i.e. $\mu$ is absolutely continuous on boundary. Is it ...
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### If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.

Please help me with this problem! Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$. If ...
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### Counterexample for a non-measurable function?

I am struggling to solve an exercise in my measure theory book and any help for solving it would be appreciated: Let $(\Omega,\mathcal{A},\mu)$ be a measure space and let $f:\Omega \to \mathbb{R}$ ...
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### Finite a.e. assumption of Egorov

I am looking for an example to show that the requirement that $f$ be finite a.e. in Egorov's theorem cannot be dropped. I was thinking about $f_n = n$, but here I am not able to see why $f_n$ does ...
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### Is there a dense subset of [0,1] of measure 1/2 whose complement is also dense?

I want to find a set $A \subset [0,1]$ so that: $A$ is dense in $[0,1]$ $A^c$ is dense in $[0,1]$ $A$ is Lesbesgue measure $1/2$ (Failing this....I want both sets to be positive measure) My first ...
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### Corollary of Lebesgue decomposition theorem and counter-example

Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if ...
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### Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?

On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties: $m(E)$ is defined for each subset $E$ of ...
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### Does the closure of an open set $O$ have the same measure as $O$? [duplicate]

Possible Duplicate: Comparing the Lebesgue measure of an open set and its closure I mean the Lebesgue measure. And one might first look at $\mathbb R^d$ before the an abstract space. ...
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### False beliefs about Lebesgue measure on $\mathbb{R}$

I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
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### An example of a family of atomic measures whose sum is not atomic

I look for a example of family of atomic measures such that their sum is not atomic. A measure $\mu$ on a $\sigma$-algebra $S$ of subsets of $X$ is called atomic if every measurable set of positive ...
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### Weird measurable set

In the following, consider the Lebegue measure in $\mathbb{R}^d$. Consider $E\subseteq \mathbb{R}^d$ measurable, with $0\lt m(E)\lt\infty$, such that any measurable subset $F$ of $E$ satisfies ...