Measurable Definition and 131 Threads

Is it true that if f:\mathbb{R}\rightarrow\mathbb{R} is a measurable function and E\subset\mathbb{R} is measurable, then f(E) is measurable? What if f is assumed to be continuous? I think that the answer is no for the first and yes for the second, but I have no idea how to prove/disprove either.

Another analysis review question: Suppose that f:\mathbb{R}\rightarrow\mathbb{R} is a measurable function and that g:\mathbb{R}\rightarrow\mathbb{R} is a Borel (i.e. Borel measurable) function. Show that f\circ g is measurable. If we only assume that g is measurable, is it still true that...

How is this possible if quarks are point particles?

I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.

Homework Statement Ω=ℝ A=σ({x}:x\in ℝ}) Determine H_{+}(Ω,A) and S_{+}(Ω,A) Homework Equations H_{+}(Ω,A) is the set of f:Ω→[0,∞) such that f is A/Borel(ℝ) measurable S_{+}(Ω,A) is the set of function in H_{+}(Ω,A) such that number of f(Ω) is finite and f(Ω) \subseteq [0,∞)...

I'm seeing the term "measurable sets" used in the definition of some concepts. But when comparing with other concepts that rely on "closed sets", I can't seem to easily find whether measureable sets are open or closed. Does anyone have any insight into that? Thanks.

Let f be a measurable nonegative function on a positive measure space,such that for every positive t, μ{x:f(x)≥t}≤M/(t^5) M is constant.prove that f is in the space L3(dμ)

What is the smallest measurable length and amount of time that can be achieved with todays technology ?

Homework Statement I have a lot of questions that ask me to prove certain functions are measureable. For example I have to show that given f:X→ ℝ is M - measurable and g:Y→ ℝ is N - measurable implies that fg is M×N measurable. Another is prove that f = {1 when x=y, 0 else} is...

Homework Statement Let {fn} be a sequence of measurable functions defined on a measurable set E. Define E0 to be the set of points x in E at which {fn(x)} converges. Is the set E0 measurable? Homework Equations Proposition 2: Let the function f be defined on a measurable set E...

Hi, All: I am trying to find a construction of a measurable subset that is not Borel, and ask for a ref. in this argument ( see the ***) used to show the existence of such sets: i) Every set of outer measure 0 is measurable, since: 0=m* (S)≥m*(S) , forcing equality. ii) Every...

Hi, All: If {f_n}:ℝ→ℝ are measurable and f_n-->f pointwise, then convergence is a.e. uniform. Are there any conditions we can add to have f_n-->f in some open interval (a,b)? Correction: convergence happens in some subset of finite measure; otherwise above not true.

1. Homework Statement [/b] Let f:ℝ\rightarrowℝ be measureable and A_{k}=\left\{x\inℝ:2^{k-1}<\left|f(x)\right|≤2^{k}\right\}, k\in \mathbb{Z}. Show that f is integrable only if \sum^{∞}_{k=-∞}2^{k}m(A_{k}) < ∞ . Homework Equations By the definition f is integrable in ℝ if and only if...

Homework Statement This should be very simple: Given the following (boundary frequency for photoelectric effect): \nu = \frac{\phi}{h} what would be the error on \nu? Homework Equations The Attempt at a Solution \varphi and h are both determined through linear regression (y = mx + c). Where...

Hello everyone! Welcome to the inaugural POTW for Graduate Students. My purpose for setting this up is to get some of our more advanced members to participate in our POTWs (I didn't want them to feel like they were left out or anything like that (Smile)). As with the POTWs for the...

Homework Statement Let f: X->R be measurable, prove that Z={(x,y)|y=f(x)} is a measurable set of XxR. Homework Equations A subset Z of XxR is measurable iff Z is a countable union of product of measurable sets of X and R. The Attempt at a Solution Let R=\cup_kV_k, where V_k are...

Homework Statement Prove that the characteristic function \chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A, where A is a measurable set of the measurable space (X,\psi) , is measurable. Homework Equations a function f: X->R is measurable if for any usual measurable set...

Given a totally finite measure μ defined on a \sigma-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [A_\alpha] where...

Hi, why is the wavefunction not measurable as it is, but is measurable when the square of the absolute value is taken? Thank you

I tried very long time to show that For closed subset A,B of R^d, A+B is measurable. A little bit of hint says that it's better to show that A+B is F-simga set... It seems also difficult for me as well... Could you give some ideas for problems?

Is it true that in GR the gauge is described by Guv while the potential is the Christoffel symbols just like the gauge in EM is described by phase and the potential by the electric and magnetic scalar and vector potential and the observable the electromagnetic field and the Ricci curvature...

The limit of an "almost uniformly Cauchy" sequence of measurable functions I'm trying to understand the proof of theorem 2.4.3 in Friedman. I don't understand why f must be measurable. The "first part" of the corollary he's referring to says nothing more than that a pointwise limit of a...

Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...

Homework Statement Prove that a set consisting of a single point is measurable and has zero area. Homework Equations Area Axioms: M is a class of measurable sets. (a) Every rectangle R \in M . If the edges of R have lengths h and k, then the area a(R) = hk . Additionally, a...

Dear All, It sounds a strange question, we know that the measure theory is the modern theory while the topological spaces is the classical analysis (roughly speaking). And measure theory solves some problems in the classical analysis. My first question is that right? Second, Is every...

Homework Statement Prove if $f$ is measurable on R and C is any closed set, f^{-1}(C) is measurable. Homework Equations Definition of measurability, closed sets etc. The Attempt at a Solution I've been trying for a while to get this proof, but I seem to just end up stuck at the...

Hi! I have a guess. Could you give me your opinion about my guess?? Let A be a rectifiable set(or jordan measurable set).This is defined in a book "Analysis on manifolds" by munkres. You can refer to it in p.112-113. Now, let f be a bounded function over the set A, and suppose f is...

Homework Statement I'm new to Measure Theory, and to be honest, I'm having a really hard time making any sense of it at all. My prof is a nice guy, but his approach to teaching involves giving zero worked solutions. This doesn't work for me. Personally I need to see solutions to get an...

Need help with this - just had this on a test and this is driving me crazy - PLEASE HELP! Let {f_{n}} be a sequence of MEASURABLE real valued functions. Prove that there exists a sequence of positive real numbers {c_{n}} such that \sum c_{n}f_{n} converges for almost every x \in \Re How is it...

Homework Statement Prove: If A is \lambda ^* -measurable and x\in \mathbb{R} ^n then x+A is \lambda ^* -measurable. My attempt at the proof is below, but i feel like it is not a correct proof. Homework Equations Notation: \lambda ^* is the lebesgue outer measure The Attempt at...

A function f: E -> \mathbb{R} is called Borel measurable if for all \alpha \in \mathbb{R} the set \{x \in R : f(x) > \alpha \} is a Borel set. If f is a strictly increasing function, then f is Borel measurable. Proof: Let H=\{x \in \mathbb{R} : f(x) > \alpha \}. I want to show that...

Homework Statement Prove that every right triangular region is measurable because it can be obtained as the intersection of two rectangles. Prove that every triangular region is measurable and its area is one half the product of its base and altitude. (Apostol's Calculus Vol1.- 1.7 Exercises)...

In Elias Stein's book Real Analysis, a measurable set E is a set such that for every \epsilon>0, there exists an open \mathscr O with the property that m_*(\mathscr{O}-E) < \epsilon. But for every open set that covers the rationals in, say, [0,1] must cover the entire interval so that the set...

As well known, for any topological space (X,T), there is a smallest measurable space (X,M) such that T\subset M. We say that (X,M) is generated by (X,T). Right now, I was wondering whether the "reverse" is true: for any measurable space (X,M), there exists a finest topological space (X,T) such...

If f : Rn -> R is Lebesgue measurable on Rn, prove that the function F : Rn * Rn -> R defined by F(x, y) = f(x - y) is Lebesgue measurable on Rn * Rn. how can I prove this question?

Homework Statement If f : Rn -> R is Lebesgue measurable on Rn, prove that the function F : Rn * Rn -> R defined by F(x, y) = f(x - y) is Lebesgue measurable on Rn * Rn. Homework Equations The Attempt at a Solution I am confused by the expression of F(x,y), it seems x-y is...

Hello, If I put a voltmeter over a (pn-junction) diode, do I measure anything? I would intuitively say "no". Is the following picture correct? So let's say the P-region is to the right, N-region to the left. If I were to attach a voltmeter across it, I'd have to attach a metal wire...

Homework Statement Prove that E is measurable if and only if E \bigcap K is measurable for every compact set K. Homework Equations E is measurable if for each \epsilon < 0 we can find a closed set F and an open set G with F \subset E \subset G such that m*(G\F) < \epsilon. Corollary...

If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable for weaker assumptions than the existence of a...

Homework Statement The question is from Stein, "Analysis 2", Chapter 1, Problem 5: Suppose E is measurable with m(E) < ∞, and E = E1 ∪ E2, E1 ∩ E2 = ∅. Prove: a) If m(E) = m∗(E1) + m∗(E2), then E1 and E2 are measurable. b) In particular, if E ⊂ Q, where Q is a finite cube, then...

Is measurable physics based on three things? Mass, the frequency of that mass, and the linear velocity that we consider measurable creation? Linear velocity is defined as something that is moving between some version of zero velocity (a hard black hole, Stephen Hawking's math) and light speed.

Problem. Let E be the closed unit square. Prove that every open subset of E is measurable. I know that one way to show that a set, say A, is measurable is to show that its outer and inner measure coincide; another way is to exibit an elementary set B such that \mu(A\Delta B)< \epsilon...

Show, that Y(x(B)) = xY(B) (Y is Lebesgue_measure ) for every borel set B and x>0. Show that also for measurable sets. I don't know how to prove anything for neither borelian or measurable sets, so I'm asking someone for doing this problem, so i can do other problems with borelian and...

hello let E,F be subset of R and a in R . show that If E is Lebesgue measurable, then E+a is Lebesgue measurable ?

let \mu^{}* , v^{}* outer measura on X . Show that max{\mu^{}* , v^{}*} is an outer measure on X ?

Homework Statement Let A \subseteq R be a Lebesgue-Measurable set. Prove that if the Lebesgue measure of A is less than infinity , then the function f(x) = \lambda(A \cap (-\infty,x)) is continous. Homework Equations The Attempt at a Solution I'm really confused about the definition of...

I've encountered two definitions of measurable functions. First, the abstract one: function f: (X, \mathcal{F}) \to (Y, \mathcal{G}), where \mathcal{F} and \mathcal{G} are \sigma-algebras respect to some measure, is measurable if for each A \in \mathcal{G}, f^{-1}(A) \in \mathcal{F}. The...

This is in the context of a homework problem but not directly related. If Mn is the collection of measurable sets of Rn under Lebesgue measure, what would be the first step in showing that M1 x M1 ⊆ M2. I'm quite convinced it's true, but my knowledge of and ability to work with the Lebesgue...

Homework Statement If for each \epsilon>0 , there is ameasurable subset E1 of E that has finite measure and a \delta>0 such that for each measurable subset A of E and index n if m(A\capE1) < \delta , then \int | fn| <\epsilon ( integration over A) Show that {fn} is tight...

I want to prove the following. Statement: Given that f is measurable, let B = {y \in ℝ : μ{f^(-1)(y)} > 0}. I want to prove that B is a countable set. (to clarify the f^(-1)(y) is the inverse image of y; also μ stands for measure) Please set me in the right direction. I would greatly...