Almost sure convergence cannot imply convergence in $L^p$. However I can't find a rigorous counterexample?
Can anybody help me? Any helpful ideas would be greatly appreciated!
2026-04-12 11:34:52.1775993692
Why almost sure convergence cannot imply convergence in $L^p$?
59 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in PROBABILITY-THEORY
- Is this a commonly known paradox?
- What's $P(A_1\cap A_2\cap A_3\cap A_4) $?
- Another application of the Central Limit Theorem
- proving Kochen-Stone lemma...
- Is there a contradiction in coin toss of expected / actual results?
- Sample each point with flipping coin, what is the average?
- Random variables coincide
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Determine the marginal distributions of $(T_1, T_2)$
- Convergence in distribution of a discretized random variable and generated sigma-algebras
Related Questions in CONVERGENCE-DIVERGENCE
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Conditions for the convergence of :$\cos\left( \sum_{n\geq0}{a_n}x^n\right)$
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Pointwise and uniform convergence of function series $f_n = x^n$
- studying the convergence of a series:
- Convergence in measure preserves measurability
- If $a_{1}>2$and $a_{n+1}=a_{n}^{2}-2$ then Find $\sum_{n=1}^{\infty}$ $\frac{1}{a_{1}a_{2}......a_{n}}$
- Convergence radius of power series can be derived from root and ratio test.
- Does this sequence converge? And if so to what?
- Seeking an example of Schwartz function $f$ such that $ \int_{\bf R}\left|\frac{f(x-y)}{y}\right|\ dy=\infty$
Related Questions in LP-SPACES
- Absolutely continuous functions are dense in $L^1$
- Understanding the essential range
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Showing a sequence is in $\ell^1$
- How to conclude that $\ell_\infty$ is not separable from this exercise?
- Calculating the gradient in $L^p$ space when $0<p<1$ and the uderlying set is discrete and finite
- $f_{n} \in L^{p}(X),$ such that $\lVert f_{n}-f_{n+1}\rVert_{p} \leq \frac{1}{n^2}$. Prove $f_{n}$ converges a.e.
- Find a sequence converging in distribution but not weakly
- Elementary use of Hölder inequality
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
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?
Take the unit interval with Lebesgue measure and $f_n(x)=n^\alpha\mathbf{1}_{(0,1/n)}$, where $\alpha>0$. Whatever $\alpha$ is, this sequence converges almost everywhere to $0$. Can you find $\alpha$ such that the convergence in $\mathbb L^p$ does not hold?