Intereting Posts

More on primes $p=u^2+27v^2$ and roots of unity
Singular and Sheaf Cohomology
A topological vector space with countable local base is metrizable
Limit without l'Hopital or Taylor series: $\lim\limits_{x \to 0} \frac{x\cos x- \sin x}{x^3}$
Modular Arithmetic question, possibly involving Chinese remainder theorem
Random walk on natural number
Urysohn's function on a metric space
Finding the order of the automorphism group of the abelian group of order 8.
Homotopy equivalence iff both spaces are deformation retracts
Prove uniform distribution
Symbol for the cardinality of the continuum
Counting couples having least common multiple less than a number
Product of Two Metrizable Spaces
If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
Let $\displaystyle f$ be differentiable, $\displaystyle f(x)=0$ for $|x| \geq 10 $ and $g(x)=\sum_{k \in \mathbb Z}f(x+k).$

In general, I know that it is not necessarily the case that the product of two Lebesgue integrable functions $f,g$ will be Lebesgue integrable. But I was reading in a textbook that if at least one of these functions is bounded, then their product will be Lebesgue integrable. How can we prove this statement? I’d appreciate some input on this, thanks.

- $G_\delta$ sets
- Nowhere monotonic continuous function
- Does one of $L^\infty$ and $L^p, p \in (0, \infty)$ contain the other?
- Field and Algebra
- Nested Division in the Ceiling Function
- Show that the set $\{ x \in : f(x) = g(x)\}$ is closed in $\Bbb R$.
- Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
- For a set of positive measure there is an interval in which its density is high, $\mu(E\cap I)> \rho \mu(I)$
- Prove that a subset of a separable set is itself separable
- Picard's existence theorem, successive approximations and the global solution

If $|f|\le M$ almost everywhere, then $|fg|\le M|g|$ almost everywhere hence $\displaystyle\int|fg|\le M\int|g|$ is finite because $g$ is integrable.

- Why is the Taylor expansion of $\cos$ decreasing?
- Definite Integral and Constant of Integration
- Determinant of matrix composition
- Relating the Künneth Formula to the Leray-Hirsch Theorem
- Compute: $\lim\limits_{n\to+\infty}\int\limits_{0}^1 e^{\{nx\}}x^{100}dx$
- Zero probability and impossibility
- How is the notion of adjunction of two functors usefull?
- Number of strict total orders on $N$ objects
- Values of $f(1986)$ knowing $f(1)=1$ and functional equation $f(a + b) = f(a) + f(b) – 2f(ab)$
- partial integration
- Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)
- How to write zero in the unary numeral system
- What is the distribution of gaps?
- Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?
- Change of coordinates between charts