Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
37,339
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Calculating PDF of a function of random variables via dirac delta integral
I came across some papers 1, 2 and others which use the following formula
$
p(y) = \int_{\mathcal{X}} \, d^{n}x \, \delta(F(\vec{x}) - y) \, p(\vec{x})
$
where $p(y)$ is the PDF of the variable ...
- 177
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0
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12
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Different measurability of Hilbert-space valued random variable
My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...
- 12.7k
2
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1
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26
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Pettis integral on locally convex space and seminorms
Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-...
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1
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The smallest nicest dense subset of a topological space
Let $X\not=\varnothing$ be a topological space, and $\mathcal{A}$ be a collection of nonempty dense subsets of $X$. Now $\mathcal{A}$ is a poset under $\subseteq$. Let $\mathcal{C}$ be a chain inside $...
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1
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Use of Fubini's Theorem in Papa Rudin's Holomorphic Fourier Transforms
I am starting to read on chapter 19, Holomorphic Fourier Transforms from Real and Complex Analysis by Walter Rudin. In the first page of that chapter I came across the function $$f(z) = \int_0^\infty ...
- 199
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0
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22
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Inverse image of an arbitrary Borel set $B \in \mathbb{R}$ under $f(x) = \min\{|x|, 1\}$
I am trying to go through some proofs and exercises about measure theory.
In particular, I have the following exercise to solve:
Let $f: \mathbb{R} \to \mathbb{R}$ be given by $f(x) = \min\{|x|, 1\}$....
- 1
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19
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Interchanging expectation and integration with a collection of random integrands
Suppose one has a collection of i.i.d. random functions $\{f(\cdot,t):t\in\mathbb R\}$, where we write $f$ for the common distribution. In a probability application, what I need is to interchange ...
- 3,013
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Measurable sets modulo null sets is a complete boolean algebra
On the complete boolean algebra wikipedia page, I found the following statement :
The algebra of all measurable subsets of a $\sigma$-finite measure space, modulo null sets, is a complete Boolean ...
- 4,685
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1
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"Proving" that $\pi=4$ with Hausdorff measure
There is a famous "proof" that $\pi=4$, which goes as follows: Start with a square with side-lengths $1$ and inscribe within it a circle with radius $1/2$.
Next, iteratively "fold" ...
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$\mu,\nu$ are Borel pr. measures s.t. $|\int fd\nu-\int fd\mu|<\epsilon$ for some Lipschitz $f$ then the inequality holds for some bounded $g\in C(X)$
Let $(X, d)$ be a metric space and $\mu,\nu$ be two Borel probability measures over $X$. Let $\epsilon > 0$ be fixed and suppose that
$$\left|\int_X fd\nu - \int_Xfd\mu\right| < \epsilon$$
for ...
- 768
2
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0
answers
36
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haar measures are decomposable [closed]
in real analysis book by folland chapter $11.1$ exercise $9$ come that:
if $G$ is a locally compact topological group with haar measure $\mu$, then $\mu$ is decomposable.
can anyone give me a proof?
A ...
1
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1
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What are the $\mathcal{M}_{sym}=\{E \operatorname{Lebesgue measurable and } E=-E \}$ measurable functions?
I am supposed to consider $L^2(-1,1)$ and the subspace $V= \{ u \in H : u \operatorname{is} \mathcal{M}_{sym}-measurable \}$ where $\mathcal{M}_{sym}$ is the $\sigma-$algebra generated by: $\{E \...
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$\mathbf{EDITED:}$if the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero $\mu(\{x\})=0$ [closed]
in folland-real analysis,chapter 11.1, exercise $9$ come that:
if the topology of locally compact topological group G is not discrete, then haar measure of singlton is zero
meaning that $\mu(\{x\})=...
2
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0
answers
12
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What is the relationship between measurable or continuos cross-sections?
Let $G$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $X$, and $\mu$ a Borel measure on $X$. To be sure, continuity of the action ...
- 21
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Is $ \mu(E \times F) \leq \nu(E \times F) \forall (E,F) \implies \mu(A) \leq \nu(A) \forall A \in \mathcal{E} \otimes \mathcal{F} $ true?
Let $(X,\mathcal{E})$ and $(Y,\mathcal{F})$ denote two measurable spaces and let $\mu,\nu$ denote two finite measures on $(X \times Y, \mathcal{E} \otimes \mathcal{F})$, where $\mathcal{E} \otimes \...
- 1,069
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46
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Are Haar measures semifinite? [closed]
We know that semifinite measure space $(X,\mathcal{M},\mu)$ is a measure space that for every measurable set $E\in\mathcal{M}$ with measure $\mu(E)=\infty$, there exist a measurable set $B\subseteq E$...
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Monotone Class Theorem and Stone-Weierstrass
We're talking about th eMonotone Class Theorem in my probability course and I've noticed some similarities with the Stone-Weierstrass Theorem. I'm told there's a number of different versions of the ...
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24
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Lipschitz continuity of translation in $L^p$
Let $f\in L^p(\mathbb{R})$ for $1\leq p <\infty,$ then $||f(x+h)-f(x)||_{L^p(\mathbb{R})} \rightarrow 0$ as $h\rightarrow 0$.
Furthermore, for $p=1,$ if $u\in L^1(\mathbb{R}) \cap W^{1,1}(\mathbb{R}...
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19
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equivalent definition for localizable measure space
I found two definition for localizable measure space.
$\textbf{definition(1):}$
$(X,\mu)$ is called localizable measure space if it can be
partitioned into a (possibly uncountable) family of ...
1
vote
0
answers
25
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Using Mass distribution principle to provide lower bound for the Hausdorff dimension of Cantor set
given the (ternary) Cantor set $\mathcal{C}$ it is well known that its Hausdorff dimension is given by $\dim_\mathcal{H}(\mathcal{C})=ln(2)/ln(3)$, which I am going to denote by $\alpha$. I am ...
1
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0
answers
18
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Principle value vs Lebsgue integrability
Though a large class of functions are Lebesgue integrable, for certain class of functions, say for example $f(x)=1/x$ Lebesgue integral over the interval $[-a,a]$ for $a>0$ is undefined. But the ...
- 364
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56
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Equivalence of different properties
Let $E \subseteq \mathbb{R}$. Show that the following are equivalent:
(i) $E$ is Lebesgue measurable ;
(ii) $|I| \geq m(I \cap E)+m(I \backslash E)$ forall interval $I \subset \mathbb{R}$ of finite ...
- 349
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1
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56
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$B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. $f$ is continuous at every element of $B$ except for a countable subset of $B$.
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 22 on p.39 in Exercises 2B in this book.
Exercise 22
Suppose $B\subset\mathbb{R}$...
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Converse to Vitali's Convergence Theorem for a subset of the sigma algebra.
The converse to Vitali's Convergence theorem, in part, states the following. Given a measure space $(X, \mathcal{A}, \mu)$, suppose that there is a sequence of real-valued $\mu$-measurable functions $\...
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Folland lemma 3.7
I am struggling to understand the following theorem's proof on Folland's Real Analysis, page 89, lemma 3.7.
Suppose that $\nu$ and $\mu$ are finite measures on $(X, \mathcal M)$. Either $\nu \perp \...
- 2,981
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1
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In what way is a Gaussian process a distribution over function space?
Gaussian processes are generally introduced as families of rvs where all finite vectors are multivariate normal. However, they are also described sometimes as "distributions over functions." ...
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Functions that can be approximated by derivatives of test functions
Let $I \subseteq \mathbb{R}$ be a compact interval. We know that functions in $L^p(I)$, $(p \geq 1)$ can be $L^1$-approximated by a sequence $(\varphi_n)_{n \in \mathbb{N}}\subseteq C_0^\infty(I)$ (...
- 5,748
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1
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Show that $T$ is a stopping time
Show that $$T = \inf \left \{n \geq 0|X_n \in \left \{0, N \right \} \right \}$$ is a stopping time with respect to $\mathcal{F}_n=\sigma(X_0,...,X_n)$ for $n\geq 0$.
I am fairly new to stopping times....
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1
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Expectation Brownian motion squared
How can I calculate
$$E_t[e^{-\frac{1}{2}\int_t^s c^2 B^2_u du -\int_t^s c B_u dB_u}B_s^2]$$
where $B_s$ is a Brownian motion?
The answer should (?) be
$$B_t e^{-2c(s-t)}+\frac{1}{2c}(1-e^{-2c(s-t)})$$...
- 1
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0
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29
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Fatou's lemma and $\limsup \int_C |f_n|^p d\mu$
Let $f_1,f_2,... ,f \in \mathscr{L}^p(X,\mathscr{A},\mu)$, where ${f_n}\to f$ a.e. and $\lim_n ||f_n||_p=||f||_p$.
We can find $A \in \mathscr{A}$ s.t. ${f_n}\to f$ uniformly with $\mu(A^c)< \...
3
votes
1
answer
46
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Existence of "induced measure" on fibers of a measurable function between measure spaces?
Let $f : X\rightarrow Y$ be a measurable function between measure spaces $X,Y$ with measures denoted $\mu,\nu$ respectively. Suppose singleton subsets of $Y$ are measurable; hence fibers of $f$ are ...
- 2,177
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73
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Show that $\int fd\mu>0$ iff $\mu(\{x\in X: f(x)>0\})>0$.
I’m practicing for my real analysis exam coming up and am specifically looking at problem 3 in section 3A from Sheldon Axler’s Measure, Integration, and Real Analysis. The question says:
Suppose $(X,...
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0
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16
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Hint for $\sigma(f^{-1}(\mathcal{C}))=f^{-1}(\sigma(\mathcal{C}))$ [duplicate]
Let $f:\Omega\rightarrow\Omega'$ and let $\mathcal{C}$ be a class of subsets of $\Omega'$. Show that
$$\sigma(f^{-1}(\mathcal{C}))=f^{-1}(\sigma(\mathcal{C}))$$ using the good set principle where, $f^{...
- 1,665
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0
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24
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Meaning of measurable set defined in terms of random variables
What is a measurable set defined in terms of random variables? I was studying Stochastic Approximation and trying to understand the proof of Dvoretzky Stochastic Approximation Theorem when this ...
- 1
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1
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61
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How can I prove this equivalent relation about measurable random variables?
How can I prove “ random variable $\alpha$ is $\mathcal{F}\vee \sigma(E_k,k=0,1,…,N)$-measurable” if and only if “$\displaystyle{\alpha=\sum_{k=0}^N} \alpha_k\mathbf{1}_{E_k},\ $ where for any k, $\...
- 21
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1
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Is union of intersections of horizontal lines and a closed set in the plane a borel set?
Let $F$ be a closed set in the plane $\mathbb{R}^2$, define $F_y=\{x\in\mathbb{R}|(x,y)\in F\}$, is $\bigcup_{y\in \mathbb{R}}F_y$ a borel set in $\mathbb{R}$? Intuitively, it is just like compressing ...
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Exercise 21 on p.39 in Exercises 2B in "Measure, Integration & Real Analysis" by Sheldon Axler. Is my proof ok?
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 21 on p.39 in Exercises 2B in this book.
Exercise 21
Prove 2.52.
2.52 condition ...
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1
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61
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Are there "measurable" properties?
Consider two measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$, consisting of sets $X$ and $Y$ and some $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$ defined on each of them respectively.
...
- 3,154
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How does one decompose a closed set into polyhedral k-chains or chainlets? Is there a decomposition or approximation theorem available?
The lecture notes of J. Harrison claim to generalize certain theorems on smooth domains, such as Stokes' theorem, to non-smooth (even fractal) domains. The theory is built on objects called polyhedral ...
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2
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77
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a function defined by inferior limit related to Borel measure is Borel measurable
Question:
$\mu$ is a Borel measure on $\mathbb R$.
Define $f:\mathbb{R\to \bar R},f(x)={\operatorname{lim inf}}_{r\to 0}{{\mu((x-r,x+r))}\over{r}}$.
Prove that $f$ is (extended) Borel measurable.
I ...
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1
answer
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+50
Half spaces are measurable
I am trying to do exercise 7.4.3. in Tao's Analysis II: Prove that the half space $E:=\{(x_1,\cdots, x_n)\in \mathbb{R}^n| x_n>0\}$ is measurable. i.e. $m^{\ast}(A)=m^{\ast}(A\cap E)+m^{\ast}(A\...
- 1,471
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2
answers
65
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A counting function that is Borel measurable
Question:
Let $F:\mathbb{R^2\to R}$ be a continuous function.
Define $p(x)$ as the number of $y$ such that $F(x,y)=0$,i.e. $p(x)=\#\{y\in\mathbb{R}|F(x,y)=0\}$.
Prove that $p(x)$ is (extended) Borel ...
- 589
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1
answer
46
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Is the inclusion $L^{\infty}L^{p} \cap L^{q}L^{\infty} \subset L^{\infty}L^{\infty}$ true?
Suppose we have the time interval $[0,T]$, a domain $\Omega = [0,1]$. I am particularly interested in the case $p=1, q=2$. In other words if we have a function
$$u \in L^{\infty}(0,T; L^{1}(\Omega)) \...
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votes
1
answer
23
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Why would $|g|$ be integrable?
For $f:(0,1)\rightarrow \mathbb{R}$ Lebesgue integrable define:
$g: [0,1]\times [0,1]\rightarrow \mathbb{R}$, such that
$$g(x,y)= \cases{\frac{f(y)}{y} \,\, \text{if}\,\, y>x \\
\\ 0 \,\, \text{...
- 226
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0
answers
118
views
+50
Measurability of a classical topological surface and its measure
Let $\Sigma \subset \mathbb{R}^3$ be a set with the following property: Given any $p\in \Sigma$, $\exists$ $W_p \subset_{\text{open}} \mathbb{R}^3$, $U_p \subset_{\text{open}} \mathbb{R}^2$ such that $...
- 356
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0
answers
19
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To show that characterization of a point mass distribution for a given function is a measure [closed]
Let $X$ be a nonempty set, and let $A$ be a $\sigma$-algebra on $X$. Let $x$ be a member of $X$. Define a function $\delta x: A \to [0,\infty]$ by letting $\delta x(A)$ be $1$ if $x \in A$ and letting ...
- 13
0
votes
1
answer
72
views
Is there more than one notion of algebras? [duplicate]
I know that an algebra is an algebraic structure, that can be seen as a vector space with a multiplication operation or as a ring with a vector space structure. However, in measure theory we define an ...
- 43
0
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0
answers
24
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Hausdorff measure of uncountable dense subsets
As far as I know, the Hausdorff measure of a countable subset is zero (please correct me otherwise). Is it possible to say the same about the uncountable dense subsets? Is there a general statement ...
- 346
-1
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0
answers
26
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Integral inequality on Bourgain Spaces
I'm studying the section about Bourgain spaces of Terence Tao's book "Nonlinear Dispersive Equations: Local and Global analysis".
I'm trying to understar the proof of Lemma 2.11 and I'm ...
- 1
0
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0
answers
13
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Reference Request: Hausdorff dimension of special Cantor sets
Currently I want to refresh my knowledge about Hausdorff dimensions and I looking for some references of the following kind:
Typical examples of calculating the dimensions are known to me but I am ...