Newbetuts

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Definition of $L^\infty$

Generalized Minkowski inequality for $L^p$ spaces

Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

Function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable

(Integral) Operator Norm: Find $||\phi||$ where $\phi : \mathcal{L^1(m)} \to \mathbb{R}$ is defined by $\phi(f) = \int (x - \frac{1}{2}) f(x) dm(x)$

Why is the undergraph definition of Lebesgue integral so rare?

On the horizontal integration of the Lebesgue integral

Bounded convergence theorem

Proof of countable additive property of Lebesgue Integrable functions

Can I derive the formula for expected value of continuous random variables from the discrete case?

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Evaluate: $\lim_{n\to\infty} \int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx.$

Converse of Jensen's inequality

Expectation of nonnegative RV

New posts in lebesgue-integral

Riemann-integrable (improperly) but not Lebesgue-integrable

Lebesgue Integral but not a Riemann integral

Differentiation under the integral sign and uniform integrability

Convergence of a series of translations of a Lebesgue integrable function

for each $\epsilon >0$ there is a $\delta >0$ such that whenever $m(A)<\delta$, $\int_A f(x)dx <\epsilon$

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Definition of $L^\infty$

Generalized Minkowski inequality for $L^p$ spaces

Computing the limit $\lim_{k \to \infty} \int_0^k x^n \left(1 - \frac{x}{k} \right)^k \mathrm{d} x$ for fixed $n \in \mathbb{N}$

Function $f$ with $|f|$ is Lebesgue integrable but $f$ isn't locally Lebesgue integrable

(Integral) Operator Norm: Find $||\phi||$ where $\phi : \mathcal{L^1(m)} \to \mathbb{R}$ is defined by $\phi(f) = \int (x - \frac{1}{2}) f(x) dm(x)$

Why is the undergraph definition of Lebesgue integral so rare?

On the horizontal integration of the Lebesgue integral

Bounded convergence theorem

Proof of countable additive property of Lebesgue Integrable functions

Can I derive the formula for expected value of continuous random variables from the discrete case?

$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.

Evaluate: $\lim_{n\to\infty} \int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx.$

Converse of Jensen's inequality

Expectation of nonnegative RV