Measurable Definition and 123 Threads

Homework Statement I have to prove that compositions of measurable mappings are measurable. i.e. If X is F/\widetilde{F} measurable and Y is \widetilde{F}/\widehat{F} measurable, then Z:=YoX:\Omega\rightarrow\widehat{\Omega} is F/\widehat{F} measurable. Homework Equations X is...

Relative to our sun, it's possible to calculate the speed at which the Earth is moving through space in it's orbit. So it's possible for an astronaut to position himself in a fixed position just slightly outside the Earths orbital path and observe the Earth passing by at the velocity the...

Hi everyone! my problem: since every simple function is bounded, we at once know, that either is our function f, cause: - \epsilon + g(x) <= f(x) <= \epsilon + g(x), so that's obviously not the problem here. this whole measure stuff doesn't get into my intuition and I don't have any...

Does the strength of a magnetic field become weaker at a constant rate the further you get away from it? I ask this question for two reasons. 1 - When iron filings are placed on a piece of paper and and a magnet is placed under it, there are definite lines of magnetic force visible that...

as constrained by the Plank Length. Any ideas on how to solve this

Homework Statement Assume S contained in R2 is bounded. Prove that if S is Riemann measurable, then so are its interior and closure 2. The attempt at a solution Proof: If S is Riemann measurable, its boundary is a zero set. Since the boundary of each open U in the int(S) is part of...

Most books have the famous example for a Lebesgue measurable set that is not Borel measurable. However, I am trying to find a Lebesgue measurable function that is not Borel measurable. Can anyone think of an example?

Homework Statement Show that if a subset E or R is measurable, then each translate E + y of E is also measurable. Relevant equations E is measurable if for all subsets A of R m^*(A) = m^*(A \cap E) + m^*(A \cap (\textbf R \setminus E)) where m* is the outer measure. The attempt...

I've been doing a little work with Borel measures and don't want to confuse Borel measurable functions with Lebesgue measurable functions for R^n -> R^m. I'm, of course, familiar with the definition that a function f:R->R is Lebesgue measurable if the preimage of intervals/open sets/closed...

Homework Statement Prove that the function $\phi(t)=t^{-1}$ is Borel measurable. Homework Equations Any measurable function into $ (\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $ \mathcal{B}(\mathbb{R})$ is the Borel sigma algebra of the real numbers $ \mathbb{R}$, is called a Borel...

Recent theoretical results http://xxx.lanl.gov/abs/0706.2522 [New J. Phys. 9 165 (2007)] http://xxx.lanl.gov/abs/0808.3324 show that weak measurements of particle velocities necessarily yield Bohmian velocities. At least, this shows that Bohmian velocities are not such an artificial...

Homework Statement Construct a measurable set E\subset [0,1] with 0<m(E)<1 such that both E and [0,1]-E (that's a set difference) are dense in [0,1]. Homework Equations None. The Attempt at a Solution Well, the obvious dichotomy here is rationals vs. irrationals, but of course the...

I'm working with caratheodory's definition of measurability of sets as given in Royden. I'm trying to prove that given any measurable set E we have that E+y={x+y|where x is contained in E} is also measurable. I'm looking for a hint not the entire solution please. I think I've actually done...

Let A be a subset of [0, 1]. And B is [0, 1] - A. Assume m_e(A) + m_e(B) = 1. Prove that A is Lebesgue measurable. m_e denotes the standard outer measure. Homework Equations A subset E of R^n is said to be lebesgue measurable, or simply measurable, if given epsilon, there exists an open set...

Homework Statement What does it mean for a function to be lebesgue measurable? What does it mean for a set to be lebesgue measureable?

This is non measurable right?

QUESTION 1: "If f\,:\,X \rightarrow \mathbb{R}^n is a function then if its coordinate functions are measurable then does this imply that f is measurable?" Suppose that we have a \sigma-algebra, \mathcal{A} defined on some set X. We say that a function f\,:\,X \rightarrow \mathbb{R}^n is...

Prove that each of the following sets is measurable, and has zero area: (a) a set consisting of a single point (b) a set consisting of a finite number of points in a plane (c) the union of a finite collection of line segments in a plane (a) To prove that a set is measurable you have to say: Let...

First I had learned this definition of a "measurable function" (Apostol): "Let I be an interval. A function f: I -> R is said to be measurable on I if there exists a sequence of step functions (s_n(x) ) such that s_n(x) -> f(x) as n -> infinity for almost all x in I." But now in other...

Lay-lines although not visible seem to be findable by some people, & points seem to line up on these lines. Churches are often built upon them. But has anyone ever measured any electrical, magnetic, or gravitational signals on these lines, or are they a complete scientific mystery.

Lets say we have the following sets: (a) set consisting of a single point (b) set consisting of finite number of points in a plane (c) union of a finite collection of line segments in a plane. We want to prove that each of these sets is measurable and has zero area. Ok so here is how I started...

Is the statement "All subsets of R are measurable" consistent with the axioms of ZF? (Specifically, of course, the omission of the axiom of choice) I've never really gotten a straight answer to this question, and I'd be very happy to hear it. :smile:

I have seen a number of definitions of measurable functions, particularly, for the following two definitions, are they the same? 1) A function f:X->R is measurable iff for every open subset T of R, the inverse image of T is measurable. 2) A function f:X->R is measurable iff for every Borel set...