Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable?
Since monotone functions are continuous away from countably many points, would that be helpful in proving the measurability?
2026-03-30 00:18:31.1774829911
Are Monotone functions Borel Measurable?
18.1k Views Asked by user48405 https://math.techqa.club/user/user48405/detail AtRelated 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 MONOTONE-FUNCTIONS
- Monotonicity of a differentiable positive function
- Convexity, Monotonicity, Positivity
- Monotonicity of function $f(x)=\sqrt[3]{(x+1)^2}-\sqrt[3]{x^2}$
- Sufficient/necessary condition for submatrix determinant (minor) that decreases with size?
- Composition of a non-increasing and a non-decreasing function
- Choosing right options based on given condition of differentiabile function
- Nowhere Monotonic/ Differentiable function proof
- Lebesgue's monotone convergence theorem, - boundedness
- Power of a decreasing sequence of positive reals.
- Does a monotone function exist such that there is a "simple" closed form for itself as well as its inverse?
Related Questions in MEASURABLE-FUNCTIONS
- Show if function is Lebesgue-measurable
- Square Integrable Functions are Measurable?
- Discontinuous Brownian Motion
- showing that $f(x)=\mu(B(x,r))$ is measurable
- Question on Durett 5.1.11
- Can someone explain the indicator function to me?
- Why the characteristic function is measurable?
- Distance metric limits to 0 if and only if convergence in measure
- Characterizing the dual space of the linear space of the signed measures generated by a given set of measures.
- $f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$
Related Questions in BOREL-SETS
- Prove an assertion for a measure $\mu$ with $\mu (A+h)=\mu (A)$
- $\sigma$-algebra generated by a subset of a set
- Are sets of point convergence of Borel functions Borel?
- Can anyone give me an example of a measurable subset of the interval [10,100], that is not a Borel set.
- If $A \subseteq \mathbb{R}$ satisfies $m^\ast(A) = 0$, then there exist $B, C ∈ \mathcal{B}(\mathbb{R})$ such that $A = B \setminus C$?
- Why is the sigma algebra generated by the set of all closed subsets a subset of the Borel sigma algebra on $\mathbb{R}$?
- Permutation of binary expansion on (0,1)
- Kernel of finitely additive function on $\mathbf{N}$ and Borel sets
- Induced Borel $\sigma$-algebra.
- Does set with Lebesgue-Mass nonzero have almost surely an open subset
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?