Newbetuts

Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7 [duplicate]

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

If $m^([-n,n] \cap E) + m^([-n,n] \setminus E) = 2n$ for all $n$, then $E$ is Lebesgue measurable

Convergence of measurable functions by two conditions

How much we can extend meaurable sets?

General property regarding outer measure for a nested sequence of sets (measurable or not).

Finding the outer measure of the x-axis in $\mathbb{R}^2$

Suppose $f_{n} \stackrel{L_{2}}{\rightarrow} f$,$g_{n} \stackrel{L_{2}}{\rightarrow} g$. Prove that $ \langle f_n,g_n \rangle\to\langle f,g\rangle $

Measurability of almost everywhere continuous functions

Is $\delta$ in $L^\infty$?

Prove convergence in $L^1$ if norms in $L^2$ are uniformly bounded

Lebesgue points of density and similar notions

Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0<\alpha<1$ an interval $I$ such that $m(A\cap I)/m(I)>\alpha$.

Show that the Cantor set is nowhere dense

Measurable subset of Vitaly set has measure zero. Proof.

Do the subsets of $\mathbb N$ that have asymptotic density form an algebra?

“Most intuitive” average of $P$ for all $x\in A \cap [a,b]$, where $A\subseteq\mathbb{R}$?

New posts in lebesgue-measure

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Show $f$ is in $L^p$ given Markov-like inequality

Compute Lebesgue measure of set of all real numbers in $[0,1]$ whose decimal representations don't contain the number 7 [duplicate]

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.

If $m^([-n,n] \cap E) + m^([-n,n] \setminus E) = 2n$ for all $n$, then $E$ is Lebesgue measurable

Convergence of measurable functions by two conditions

How much we can extend meaurable sets?

General property regarding outer measure for a nested sequence of sets (measurable or not).

Finding the outer measure of the x-axis in $\mathbb{R}^2$

Suppose $f_{n} \stackrel{L_{2}}{\rightarrow} f$,$g_{n} \stackrel{L_{2}}{\rightarrow} g$. Prove that $ \langle f_n,g_n \rangle\to\langle f,g\rangle $

Measurability of almost everywhere continuous functions

Is $\delta$ in $L^\infty$?

Prove convergence in $L^1$ if norms in $L^2$ are uniformly bounded

Lebesgue points of density and similar notions

Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0<\alpha<1$ an interval $I$ such that $m(A\cap I)/m(I)>\alpha$.

Show that the Cantor set is nowhere dense

Measurable subset of Vitaly set has measure zero. Proof.

Do the subsets of $\mathbb N$ that have asymptotic density form an algebra?

“Most intuitive” average of $P$ for all $x\in A \cap [a,b]$, where $A\subseteq\mathbb{R}$?