Intereting Posts

Holomorphic functions on a complex compact manifold are only constants
$f(x)=3x+4$ – Injective and Surjective?
Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$
Find the approximate center of a circle passing through more than three points
How can I express $\sum_{k=0}^n\binom{-1/2}{k}(-1)^k\binom{-1/2}{n-k}$ without using summations or minus signs?
What does double vertical lines $\|$ mean in number theory?
Limit of $2^{1/n}$ as $n\to\infty$ is 1
What does triple convolution actually look like?
Understanding the definition of the generator of a semigroup of operators
A necessary condition for series convergence with positive monotonically decreasing terms
A ring without the Invariant Basis Number property
An awful identity
When an ideal is generated by idempotents
How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?
$(-27)^{1/3}$ vs $\sqrt{-27}$.

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.

- Is this space complete?
- Complex Measures: Lebesgue
- If one side of $\int f\ d\lambda = \int f\ d\mu - \int f\ d\nu$ exists, does the other one exist as well?
- Questions of an exercise in Lebesgue integral
- Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?
- Generalisation of Dominated Convergence Theorem

- What is the relationship between the following set functions?
- Mollifiers: Approximation
- Are all measure zero sets measurable?
- Nonmeasureable subset of ${\mathbb{R}}^2$ such that no three points are collinear?
- lebesgue measure/Measurable sets
- Showing that if $f=g$ a.e. on a general measurable set (for $f$, $g$ continuous), it is not necessarily the case that $f=g$.
- Difference of differentiation under integral sign between Lebesgue and Riemann
- Topology of convergence in measure
- Find a non-negative function on such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable
- What is the Lebesgue measure of the set of numbers in $$ that has two thirds of ones in their infinite base-2 expansion?

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

- Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
- When is the union of topologies a topology?
- Exercise 1 pg 33 from “Algebra – T. W. Hungerford” – example for a semigroup-hom but not monoid-hom..
- What is a good graphing software?
- Derivation of weak form for variational problem
- if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$
- Derivation of inverse sine, what is wrong with this reasoning?
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- Ellipse in polar coordinates
- Dihedral subgroups of $S_4$
- Prove that a set consisting of a sequence and its limit point is closed
- How do I maximize $|t-e^z|$, for $z\in D$, the unit disk?
- Prove that $X'$ is a Banach space
- Prove that if $a\equiv b \pmod m $ , then $a \bmod m = b \bmod m$