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The following are the results from a wikipedia article about $L_p$ space:

a. Let $0 ≤ p < q ≤ ∞$. $L_q(S, μ)$ is contained in $L_p(S, μ)$ iff $S$ does not contain sets of arbitrarily large measure;

b. Let $0 ≤ p < q ≤ ∞$. $L_p(S, μ)$ is contained in $L_q(S, μ)$ iff $S$ does not contain sets of arbitrarily small non-zero measure.

Here are my **questions**:

- Can an uncountable family of positive-measure sets be such that no point belongs to uncountably many of them?
- Fast $L^{1}$ Convergence implies almost uniform convergence
- Prove $E((X+Y)^p)\leq 2^p (E(X^p)+E(Y^p))$ for nonnegative random variables $X,Y$ and $p\ge0$
- Must the (continuous) image of a null set be null?
- locally compact metric space, regular borel measure
- Why do we essentially need complete measure space?

- What does “sets of arbitrarily large measure” mean? Is “$S$ does not contain sets of arbitrarily large measure”
*equivalent*to “$\mu(S)<+\infty$”? - What does “sets of arbitrarily small non-zero measure” mean?

[Added:]

There is a result in *Another note on the inclusion $L^p(\mu) ⊂ L^q(\mu)$*(

by A. Villani, The American Mathematical Monthly, Vol. 92 (1985), No. 7, 485–487):

The following conditions on measure space are equivalent:

- $\sup_{E\in{\mathscr A}_{\infty}}\mu(E)<+\infty$
- $L^p(\mu)\subset L^q(\mu)$ for all $p,q\in(0,\infty)$ with $p>q$

where ${\mathscr A}_\infty=\{E\in\Sigma:\mu(E)<+\infty\}$.

This is similar with (a) but $p,q\in(0,+\infty)$.

- Riemann-Stieltjes integral, integration by parts (Rudin)
- If $\alpha$ is an irrational real number, why is $\alpha\mathbb{Z}+\mathbb{Z}$ dense in $\mathbb{R}$?
- Uniform convergence in an open interval of a power series
- Is there a function whose inverse is exactly the reciprocal of the function, that is $f^{-1} = \frac{1}{f}$?
- Baby Rudin: Chapter 1, Problem 6{d}. How to complete this proof?
- Infimum and supremum of the empty set
- Interpolation inequality
- Fundamental Optimization question consisting of two parts.
- $\epsilon{\rm -}\delta$ proof of the discontinuity of Dirichlet's function
- An open interval as a union of closed intervals

This is more of a comment response to Jack, but I need the formatting given by MarkDown.

The problem is that the Wikipedia sentence is ambiguous: and in one way of reading it, the statement is incorrect. Wikipedia contains:

Let $0\leq p\leq q\leq \infty$. $L^q(S,\mu)$ is contained in $L^p(S,\mu)$ iff $S$ does not contain sets of arbitrarily large measure.

A big spot where it is entirely unclear is whether the phrase “Let $0\leq \ldots$” is part of the “left-hand-side” of the “iff”. The usage of a full stop `.`

instead of a comma `,`

would suggest that it is not.

Then if we interpret “does not contain…” as what you wrote in your question, that is $\mu(S) < \infty$, then the statement is false! Because as in my comment there exists $p,q\in (0,\infty)$ and a measure space $(S,\mu)$ such that $\mu(S) = +\infty$ while the $L^p \subset L^q$ inclusion holds, contradicting the “only if” of the “iff”.

If we interpret “does not contain…” as what Villani wrote, that is $\sup_{A_\infty} \mu(E) < +\infty$, it is clear that we will have a problem with the endpoints. So *neither interpretation works*!

The only way to fix the statement is to *correctly place your quantifiers*. That is to say, you can write that

Given a measure space $(S,\mu)$, the following two conditions are equivalent:

- $\mu(S) < \infty$
- For every $0 < q \leq p \leq \infty$ the inclusion $L^p \subseteq L^q$ holds.

I do not include the endpoint $0$ as I do not know how you want to define the $L^0$ space when $\mu(S) = + \infty$. To show that $1\implies 2$ you just use Hölder/Jensen; to show that $2\implies 1$ just take the function $f\equiv 1$ which is in $L^\infty$. Integrating in any $L^p$ for $p <\infty$ tells you that $S$ has finite measure.

Pick a positive number $r$. Does $S$ have sets with measure bigger than $r$? Deos $S$ have sets of measure smaller than $r$ but not zero?

If, for every value of $r$ you answered yes to the first question, $S$ contains sets of arbitrarily large measure.

If, for every value of $r$ you answered yes to the second question, $S$ contains sets of arbitrarily small, nonzero measure.

- Some equivalent formulations of compactness of a metric space
- Krull's theorem and AC
- Proof of “the continuous image of a connected set is connected”
- Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
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