Intereting Posts

How to find the sum of the sequence $\frac{1}{1+1^2+1^4} +\frac{2}{1+2^2+2^4} +\frac{3}{1+3^2+3^4}+…$
Find a non-negative function on such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable
Do such sequences exist?
Countable product of complete metric spaces
Covariant and partial derivative commute?
how to show $f$ attains a minimum?
How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers?
Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time
Number of ways to partition a rectangle into n sub-rectangles
Repertoire Method Clarification Required ( Concrete Mathematics )
Is mathematical history written by the victors?
Expected Values of Operators in Quantum Mechanics
Subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
What do we mean when we say a differential form “descends to the quotient”?
Optimal strategy for this Nim generalisation?

I could show that $\|\cdot\|_p$ is decreasing in $p$ for $p\in (0,\infty)$ in $\mathbb{R}^n$. Following are the details.

Let $0<p<q$. We need to show that $\|x\|_p\ge \|x\|_q$, where $x\in \mathbb{R}^n$.

If $x=0$, then its obviously true. Otherwise let $y_k=|x_k|/\|x\|_q$. Then $y_k\le 1$ for all $k=1,\dots,n$. Therefore $y_k^p\ge y_k^q$, and hence $\|y\|_p\ge 1$ which implies $\|x\|_p\ge \|x\|_q$.

- The reflexivity of the product $L^p(I)\times L^p(I)$
- Relative countable weak$^{\ast}$ compactness and sequences
- What is the predual of $L^1$
- Banach Spaces: Uniform Integral vs. Riemann Integral
- show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)
- Equivalence of definitions of $C^k(\overline U)$

The same argument works even for $x\in \mathbb{R}^{\mathbb{N}}$.

I am wondering whether the result is true for functions $f$ in a general measure space $(\mathcal{X}, \mu)$. The same technique doesn’t seem to work in general.

I know that its certainly not true in the case when $\mu(\mathcal{X})<\infty$, as in this case $\|f\|_p\le \|f\|_q\cdot \mu(\mathcal{X})^{(1/p)-(1/q)}$ for $p<q$, so in particular if $\mu$ is a probability measure then, in fact, $\|f\|_p$ is increasing in $p$.

The question is the following. If $f$ is a real valued function on a measure space $(\mathcal{X}, \mu)$ and if $\|f\|_p$ is defined for all $p>0$, is there any result like $\|f\|_p$ is decreasing in $p$?

- Set of continuity points of a real function
- Measurable functions on product measures
- Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
- Two-valued measure is a Dirac measure
- Characterisation of norm convergence
- Cesàro operator is bounded for $1<p<\infty$
- Inequality between $\ell^p$-norms
- If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?
- When does intersection of measure 0 implies interior-disjointness?
- Reference book on measure theory

Sometimes we can not say anything. Consider the case

$L^p(\mathbb{R})\,$ for $p=1\,$ and $p=2\,$ and look at the characteristic function $\chi_{E}\,$ of a measurable set $E.$ Then

$$\|\chi_E\|_{L^1}=\int_E dx =m(E)$$

while

$$\|\chi_E\|_{L^2}=\left(\int_E dx\right)^{1/2}=\sqrt{m(E)}.$$ Now if $m(E)>1\,$ we have $m(E)>\sqrt{m(E)}\,$ while $m(E)<\sqrt{m(E)}\,$ in the case $0<m(E)<1$.

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