Let $E \subseteq\mathbb{R}$. Then
For all open interval $I \subseteq \mathbb{R}$ , $m^∗(I) = m^∗(I∩E)+m^∗(I∩E^c) \Rightarrow E \textrm{ is lebesgue measurable}$
I want to solve this statement. However, I can’t.
I need your advice.
2026-03-25 19:04:58.1774465498
I want to prove or disprove "∀ open interval I, m∗(I) = m∗(I∩E)+m∗(I∩E^c)" implies "E is lebesgue measurable".
107 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in MEASURE-THEORY
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Absolutely continuous functions are dense in $L^1$
- I can't undestand why $ \{x \in X : f(x) > g(x) \} = \bigcup_{r \in \mathbb{Q}}{\{x\in X : f(x) > r\}\cap\{x\in X:g(x) < r\}} $
- Trace $\sigma$-algebra of a product $\sigma$-algebra is product $\sigma$-algebra of the trace $\sigma$-algebras
- Meaning of a double integral
- Random variables coincide
- Convergence in measure preserves measurability
- Convergence in distribution of a discretized random variable and generated sigma-algebras
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- $f\in L_{p_1}\cap L_{p_2}$ implies $f\in L_{p}$ for all $p\in (p_1,p_2)$
Related Questions in LEBESGUE-MEASURE
- A sequence of absolutely continuous functions whose derivatives converge to $0$ a.e
- property of Lebesgue measure involving small intervals
- Is $L^p(\Omega)$ separable over Lebesgue measure.
- Lebesgue measure and limit of the integral.
- uncountable families of measurable sets, in particular balls
- Joint CDF of $X, Y$ dependent on $X$
- Show that $ Tf $ is continuous and measurable on a Hilbert space $H=L_2((0,\infty))$
- True or False Question on Outer measure.
- Which of the following is an outer measure?
- Prove an assertion for a measure $\mu$ with $\mu (A+h)=\mu (A)$
Related Questions in OUTER-MEASURE
- True or False Question on Outer measure.
- Which of the following is an outer measure?
- Which of the following is true about lebesgue measure
- set of points in $E_n$ infinitely often and almost always are in $\sigma$-algebra
- A countable set has outer measure zero. Explanation of Royden example.
- Finding all outer measures on a finite set
- Outer measure in a finite measure space.
- Can a compact connected set in $\mathbb{R}^n$ with empty interior have positive Lebesgue measure?
- n dimensional measure of homeomorphism of interval
- Let $\mu$ be a finite measure on $(X, M)$, and let $\mu^*$ be the outermeasure induced by $\mu$.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
https://math.stackexchange.com/a/3538939/282784
I found proof of my question. Some of you may need proof, so I leave the link here.