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Followings are from Wikipedia.

  1. Why is it that:

    Colloquially, if $1 ≤ p < q ≤ ∞$,

    • $L^p(S, μ)$ contains functions that are more locally singular,
    • while elements of $L^q(S, μ)$ can be more spread out?

    I was also wondering what "locally singular" and "spread out" mean mathematically?

  2. Why is it that:

    Consider the Lebesgue measure on the half line $(0, ∞)$.

    • A continuous function in $L^1$ might blow up near 0 but must decay sufficiently fast toward infinity.
    • On the other hand, continuous functions in $L^∞$ need not decay at all but no blow-up is allowed?

    I was also wondering what "blow up (near 0)" and "decay sufficiently fast (toward infinity)" mean mathematically?

  3. Although it is stated in the following, I don't understand how this "precise technical result" related to the above two quotes?

    The precise technical result is the following:

    • 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, and
    • 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.
Thanks and regards!
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    I believe this approach is far too abstract. As I suggested in a previous question of yours: do you have some toy examples of functions which belong to $L^1[0,1]$, but not $L^2[0,1]$ or $L^2[0,1]$ but not $L^\infty[0,1]$? Do you have some examples of sequences in $\ell^2$, but not in $\ell^1$ or $\ell^\infty$, but not $\ell^2$? If not, find them! If so, try to understand their features and how they meet the descriptions in your post. I would strongly recommend to answer these explicit questions on the simplest of measure spaces, i.e., $[0,1]$ and $\mathbb{N}$, first.2012-12-28
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    @Martin: Thanks! http://math.stackexchange.com/a/18399/12812012-12-28

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