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mensurable

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mensurable
mensurable, a. (ˈmɛnsjʊərəb(ə)l, mɛnʃ(j)ʊər-) [a. F. mensurable, ad. late L. mēnsūrābil-is, f. mēnsūrāre to measure, f. mēnsūra measure n.: see -able.] 1. Capable of being measured; hence, having assigned limits.1604 T. Wright Clymact. Years 5 Loe thou hast put my dayes mensurable. 1694 Holder Disc.... Oxford English Dictionary
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immensurable
immensurable, a. (ɪˈmɛnsjʊərəb(ə)l, -ʃər-) [a. F. immensurable (15th c. in Godef.), or ad. late L. immensūrābilis (5th c.), f. im- (im-2) + mensū rābilis mensurable.] Immeasurable.1535 Stewart Cron. Scot. II. 426 In meit and drink, and sleip also wes he Immensurabill and out of temperance. 1612 W. P... Oxford English Dictionary
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Petrus de Cruce
Franconian notation, so called after the theorist Franco of Cologne, who outlined the system in his c. 1260 treatise, Ars cantus mensurabilis (The art of mensurable wikipedia.org
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unmensurable
unˈmensurable, a. (un-1 7 b.)1513 Douglas æneid x. Prol. 93 Consider quhou he [sc. God] is onmensurabyll; Him, as he is, to knaw thou art not abyll. 1683 Cave Ecclesiastici, Athanasius 47 This he cryes out upon as..an unmensurable madness. Oxford English Dictionary
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mensurability
mensurability rare. (mɛnsjʊərəˈbɪlɪtɪ, mɛnʃ(j)ʊər-) [f. mensurable + -ity.] The state or quality of being mensurable.1678 Cudworth Intell. Syst. 66 Whatsoever is, is Extended or hath Geometrical Quantity and Mensurability in it. 1814 D. Stewart Philos. Hum. Mind Note (G) II. 511 The common quality w... Oxford English Dictionary
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mensural
mensural, a. (ˈmɛnsjʊərəl, ˈmɛnʃ(j)ʊərəl) [a. med.L. mensūrāl-is, f. L. mēnsūra measure n.: see -al1.] 1. Pertaining to measure.1651 Wittie tr. Primrose's Pop. Err. iv. xxv. 301 There were among the Ancients as well pounds in measure, as in weight, for their vessels were drawn about with lines..and ... Oxford English Dictionary
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Let $I=[0, \infty)$ and $f:I \to R$ a mensurable function such that $|f(t)| \leq \frac{t^\alpha}{1+t}$, where $0 <\alpha <1$. Let $I=[0, \infty)$ and $f:I \to \mathbb{R}$ be a mensurable function such that $|f(t)| \le...
Just calculate: $\int_I{\int_I{e^{-tx}\frac{t^\alpha}{1+t}dx}dt} = \int_I{\frac{t^\alpha}{1+t}\int_I{e^{-tx}dx}dt} = \int_I{\frac{t^\alpha}{1+t}\frac{1}{t}dt} 1, 1+t>t$ respectively for two divided interval $(0,1)$ and $(1,\infty)$ for $t.$
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dimensuration
† dimensuˈration Obs. [n. of action f. L. type *dīmensūrāre, f. dī- + mensūrāre to measure, after dīmetīrī, dīmensus, f. di- + metīrī, mensus to measure.] Measuring out or off, measurement.1593 Norden Spec. Brit., M'sex i. Prepar. 15 Such an expected geographicall description..doeth require dimensur... Oxford English Dictionary
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About integrable functions. Let $f_n\colon [0,1]\rightarrow R$ be Lebesgue mensurable with $\int_{0}^{1} |f_n(t)|^3dm(t)<1$ for all $n$. How we can show that $f_n$ is integrable uniformly i.e for all $\epsilon>0$ ther...
We can use Hölder inequality for $p= 3$ and $q=\frac 32$. Then we have for $n\in\mathbb N$ and $E\subset [0,1]$ measurable $$\int_E |f_n(t)|dm(t)=\int_{[0,1]} |f_n(t)|\mathbf 1_E(t)dm(t)\leq \left(\int_{[0,1]}|f_n(t)|^3dm(t)\right)^{1/3}\mu(E)^{2/3}\leq \mu(E)^{2/3}$$ so for a fixed $\varepsilon$, t...
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longa
‖ longa Mus. (ˈloŋga) Also 7 longo. [It., a. med.L. longa (sc. nota), fem. of longus long.] = long n. 4.c 1648–50 R. Brathwait Barnabees Jrnl. (1818) 181 What though brieves too be made longo's? 1753 in Chambers Cycl. Supp. 1893 Shedlock tr. Riemann's Dict. Mus., Longa ({duplong2}), the second longe... Oxford English Dictionary
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Let $C \in \mathbb{R}^n$ be a Jordan mensurable set of measure zero. Show that $\int_A \chi_C = 0$, for every closed rectangle $A \supset C$. Let $C \in \mathbb{R}^n$ be a Jordan mensurable set of measure zero. Show t...
We can write A as $C\cup (A - C)$ and we can express the integral as $\int_A \chi_{C} = \int_C \chi_{C} \ + \int_{A-C}\chi_{C}$. By the definition of $\chi_{C}$, we have that $\chi_{C}=0$ on $A-C$, so the second integral of the sum is zero. Then $\int_A \chi_{C} = \int_C \chi_{C} = v(C)$. Which is z...
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Notation and a problem with Aleatory variables (Advanced Probability) I am studying advanced probability and I have a question with notation. One exercise says: Let $(\Omega,B)$, show that $A \in B$ iff $1_A \in B$. B...
As zoli answered above $1_A \in \mathcal{B}$ means that $1_A$ is a $\mathcal{B}$- measurable function. I would like to address your second question: If $F(x)$ is continuous, then it is clear that $F$ is measurable, but let's look at Y = F(X): for $a > 0$, the set $\\{x: F(x)<a\\} = (-\infty, Y(a)) $...
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measured
measured, ppl. a. (ˈmɛʒ(j)ʊəd) [f. measure n. and v. + -ed.] † 1. Moderate, temperate. (Also well measured.)c 1400 Rule St. Benet 2328 A Priores..Aw to be mesured euermair To bind non als bot þai may bere. 1456 Sir G. Haye Law Arms (S.T.S.) 300 To be wele mesurit in..eting and drinking. 1483 Caxton ... Oxford English Dictionary
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Measure Theory and Cantor Set (couterexample). > Consider $O_{n} = \\{y \in \mathbb{R}^{d} \mid \exists x \in E \text{ with } |y - x| < \frac{1}{n}\\}$ where $E$ is a mensurable set. > > (a) Prove that if $E$ is comp...
What you need is a fat Cantor set, since the original Cantor set has measure zero and your strict inequality cannot be attained. Let $C_F \subset [0,1]$ be a fat Cantor set and let $E = [0,1]\setminus C_F$. Then for all $n$ we have $[0,1] \subset O_n$: let $x\in [0,1]$. If $x\in C_F$ then $x\in O_n$...
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$\{(a,b) \in A \times B : f(a)=g(b)\} \in \mathcal A \otimes \mathcal B$ Let $(A,\mathcal A), (B, \mathcal B), (C, \mathcal C)$ be measure spaces. Let $f:A\rightarrow C$ and $g :B\rightarrow C$ be mensurable functions...
Since $D \in \mathcal C \otimes \mathcal C,$ it's easy to see that if we define $h:A\times B\rightarrow C \times C $ by $h(a,b)=(f(a),g(b))$, then $h$ is mensurable and since $h^{-1}(D)=E$ and $D$ is mensurable we get that $E$ is mensurable, that is, $E \in \mathcal A \otimes \mathcal B$.
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