# Questions tagged [outer-measure]

Outer measure on $X$ is a function $\phi : 2^X \rightarrow [0,\infty]$ defined on all subsets on $X$ that satisfies the following : (1) $\phi ( \emptyset )=0$ (2) Monotonicity : $A\subset B$ implies $\phi (A)\leq\phi (B)$ and (3) Countable subadditivity : $\phi (\bigcup_{i=1}^\infty A_j) \leq \sum_{j=1}^\infty \phi (A_j)$

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### Volume of an interval in $R^N$ seen as the supremum of set of volumes of compact subintervals

In a Lemma to be used in developing properties of Lebesgue Outer Measure on $R^N$ ($N$ is a positive integer) the professor pointed that it is not hard to establish that the volume of an interval $I$ ...
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### Measure Theory question about Outer measure vs measure

We are currently covering the last chapter of baby Rudin where he quickly covers the basics of Measure Theory. I am having a hard time understanding under what circumstances does the outer measure ...
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### Are these two families of sets the same without the assumption of measurability?

Suppose we have a nonempty set $X$ and an outer measure $\mu$ on $\mathcal{P}(X)$. Let's denote the $\sigma$-algebra of $\mu$-measurable sets, i.e. $\sigma$-algebra from Caratheodoty extension theorem,...
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### An interesting qn came in my mind related to outer measure of the cartesian product of A and B??

So here are the questions came in my mind, but I could not answer each of them.. If $A$ and $B$ are two subsets of $\Bbb{R}$ then $\lambda_2^*(A×B)=\lambda_1^*(A) \lambda_1^*(B)$?? (If not the case, ...
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### For any set $E\subset\mathbb{R}^{p+q}$, if for all $x\in\mathbb{R}^p$ that $E(x)$ is measurable, does $E$ measurable?

The definition $E(x)=\left\{y\big|y\in\mathbb{R}^q\land <x,y>\in E\right\}$ is all the points that in E while $x$ is selected. And furthermore, $E$'s projection in $\mathbb{R}^p$ is also an ...
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### 1.3.9 from the book measure theory of Donald L. Cohn

I've read the solution of this exercize from this link Cohn Measure Theory exercise question 9 chapter 1.3 other direction When I've tried to prove that $\lambda^* ( B_1 - B_0 ) = 0$ with the hint ...
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### Show that induced outer measure of the sum of two measures is the sum of the induced outer measures

Let $H$ be a semiring over $\Omega$ and let $\mu$ and $\nu$ be two measures on $H$. Show that $(\mu + \nu)^* = \mu^* + \nu^*$ with the induced outer measures $\mu^*$ and $\nu^*.$ We defined the ...
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### Outer measure "distributive" property

Let $X$ be a set, $\mathcal{A}\subseteq 2^X$ is a ring of sets of $X$, $\mu: \mathcal{A}\to [0, \infty]$ is a countably additive function. And let $\mu^*$ is outer measure induced by $\mu$ on $2^X$. ...
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### Prove $E$ is lebesgue measurable set and calculate lebesgue measure of E

Can someone help me with the following problem? Let $E$ to be to set of all real numbers in the interval $[0,1]$ which in thier decimal expension the digit $"1"$ appears only in a finite number of ...
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### Compact interval $I=[a,b]\subseteq \mathbb{R}$ is an $\mathfrak{L}$-measurable set. [closed]

Let $\mathfrak{L}$ denote the Lebesgue (outer) measure on $\mathbb{R}$. Show, directly using the definition of $\mathfrak{L}$, that a compact interval $I=[a,b]\subseteq \mathbb{R}$ is a $\mathfrak{L}$-...
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### Understanding when a set is outer-measurable

Suppose $\mu$ is an outer measure on some arbitrary set $X$. We say a subset $E$ of $X$ is $\mu$-measurable if and only if $$\mu(A) = \mu(A \cap E) + \mu(A \setminus E), \quad \forall A\subset X.$$ I ...
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### Lebesgue Outer Measure Additivity for disjoint intervals.

I am trying to prove that if $\cup_{i=1}^{\infty}I_i$ are disjoint open intervals then $$|\cup_{i=1}^{\infty}I_i|=\sum_{i=1}^{\infty}\mathcal{l}(I_i).$$ Where $||$ is the Lebesgue outer measure. I ...
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### Lebesgue Outer Measure Null Set Proof

I am trying to solve the following proof: Let $||$ be the Lebesgue outer measure. Show that if $|B|=0$ then $|A\cup B|=|A|$. If I consider $B=\{x_1,x_2,....\}$ this is easy enough. Consider an ...
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### For bounded $A \subset \mathbb{R}$ we define: $m_*(A) := m(E)-m^*(E\setminus A), E\subset R,A\subset E, m(E)<\infty$ show that $m_*$ is well defined

I was trying to solve the following question: For bounded $A \subset \mathbb{R}$ we define: $$m_*(A) := m(E)-m^*(E\setminus A)$$ For Lebesgue measurable set $E \subset \mathbb{R}$, where $A\subset E$ ...
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### If $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is open, $|A \cup G| = |A| + |G|$

I attached the book link here, you may check the proof on Page 47. Additivity of outer measure if one of the sets is open. Suppose $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is open. ...
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### Prove $\mu^*(A)=\nu(A)$ if there exists a cover $A\subset \cup_{n\geq1} B_n$ and $\mu^*(B_n)<\infty \;\forall n\geq 1$

Given $\mu : \mathscr{H} \to \mathbb{R}$ a pre-measure on $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$, $\mathscr{A}=\sigma(\mathscr{H})$ and $\nu$ a measure such ...
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### Extension of measure is less or equal to outer measure when restricted to semiring [closed]

If $\mu : \mathscr{H} \to \mathbb{R}$ is a pre-measure on $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$, $\mathscr{A}=\sigma(\mathscr{H})$ and $\nu$ a measure such ...
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### Inclusion-Exclusion Principle for Outer Measure

Question: Let $\mu^*$ be an outer measure on a set $\Omega$ and $E$ be a $\mu^*$-measurable set. Show that $$\mu^*(A) + \mu^*(E) = \mu^*(A \cap E) + \mu^*(A \cup E)$$ for all $A \subseteq \Omega$. ...
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### Outer Measure and $\sigma$-algebra on $\Omega = \Big\{(i, j)\ \big|\ 1\le i,j \le N\Big\}$

Question: Let $N\in\mathbb{N}$ and $$\Omega = \Big\{(i, j)\ \big|\ 1\le i,j \le N\Big\}\subseteq \mathbb{N}\times\mathbb{N}$$ Define $\mu^*: \mathcal{P}(\Omega) \to [0, \infty]$ by letting $\mu^*(A)$...
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### Show that the collection of sets for which the inner measure equals the outer measure $\mu_*(A) = \mu^*(A)$ is a $\sigma$-algebra.
On space $\Omega$ we have algebra $\mathcal{A} \subset \mathcal{P}(\Omega)$ with measure $\mu: \mathcal{A} \to [0,1]$ and we define the inner measure $\mu_*: \mathcal{P}(\Omega) \to [0,1]$ and outer ...