Intereting Posts

Can all rings with 1 be represented as a $n \times n$ matrix? where $n>1$.
Why does $\mathrm{ord}_p(n!)=\sum_{i=1}^k a_i(1+p+\cdots+p^{i-1})$?.
Evaluating $\lim\limits_{n \to \infty }\sqrt{ \frac{\left | \sin1 \right |}{1}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}} $
determine the closures of the set k={1/n| n is a positive integer}
Why use the derivative and not the symmetric derivative?
When can the maximal sigma algebra be generated by all singleton subsets?
$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$
$L^{2}(0,T; L^{2}(\Omega))=L^{2}(\times\Omega)$?
Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real
Property of $\dfrac{\sum a_i}{\sum b_i}$ when $\dfrac{a_i}{b_i}$ is increasing
What is a short exact sequence telling me?
Good resources (book or otherwise) to learn/study basic Combinatorics
Closed form for $\sum\limits_{k=1}^{\infty}\zeta(4k-2)-\zeta(4k)$
Surface with non-zero mean curvature means orientable
How to find a measurable but not integrable function or a positive integrable function?

Let $f$ be a Lebesgue integrable function on $[0,2]$. If $\int_E fdx=0$ for all measurable set $E$, such that $m(E)=\pi/2$. Is $f=0$ a.e. Prove or disprove

I could not figure out anything. Can a function be very oscillatory so that on every interval its integral is zero?

Any hint and approach are welcome.

- Lebesgue integration calculation problem?
- Prove that lebesgue integrable equal lebesgue measure
- Is there a general theory of the “improper” Lebesgue integral?
- Calculate the Riemann Stieltjes integral
- Limit of the integral $\int_0^1\frac{n\cos x}{1+x^2n^{3/2}}\,dx$
- Integral in $L^p$ spaces

- Generalized convex combination over a Banach space
- Deduce that $f=0$ a.e.
- Can a function that has uncountable many points of discontinuity be integrable?
- Show that if $E\subset\mathbb{R}$ is a measurable set, so $f:E\rightarrow \mathbb{R}$ is a measurable function.
- lebesgue measure/Measurable sets
- measurability with zero measure
- Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.
- $L^p$ and $L^q$ space inclusion
- Prove that, there exists no continuous function $f:\mathbb R\rightarrow\mathbb R$ with $f=\chi_{}$ almost everywhere.
- If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise

Hint: let $A$ and $B$ be small enough (let’s say $m(A)=m(B)<0.001$). Then there exists $C$ disjoint from $A \cup B$ such that $m(C)=\pi/2 – m(A)$. Using your conditions on $A \cup C$ and $B \cup C$ show that $\int_Afdm=\int_Bfdm$. Now consider the positivity and negativity sets of $f$. Show, that if one or both of them have measure zero you are done. Show that otherwise it will contradict previous argument

- No group of order 36 is simple
- How to get the characteristic equation?
- Why is the localization at a prime ideal a local ring?
- Show that there is a step function $g$ over $$
- Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$
- Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$
- Wild automorphisms of the complex numbers
- Prove the general arithmetic-geometric mean inequality
- Prove that, for any natural number $k$ $\exists m \in M $ s.t i) $m$ has exactly $k$ $1$'s, an
- Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
- Number of ways of forming 4 letter words using the letters of the word RAMANA
- Turning a Piecewise Function into a Single Continuous Expression
- Can we show that the decimal expansion of $\pi$ doesn't occur in the decimal expansion of the Champernowne constant?
- Gambling puzzle
- covering map with finite fibres and preimage of a compact set