I am trying to understand how Holder's inequality applies to the counting measure. The statement of Holder's inequality is:

Let $(S,\Sigma,\mu)$ be a measure space, let $p,q \in [1,\infty]$ with $1/p + 1/q = 1$. Then for all measurable, real-valued functions $f$ and $g$ on $S$:

$$ \lVert fg\rVert_1 = \lVert{f}\rVert_p \lVert{g}\rVert_q.$$

The norm here is given by $\lVert{f}\rVert_p = \left(\int_S\mid{f}|^p|d\mu\right)^{1/p}$.

Case 1: For Lebesegue measure $\mu$ with $S = \mathbb{R}^n$, Holder's inequality becomes:

$$ \int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x\biggr)^{\frac{1}{p}} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\frac{1}{q}}. $$

Here, the functions $f$ we are integrating have signature $f: \mathbb{R}^n \to \mathbb{R}$. The symbol $x \in \mathbb{R}^n$ here denotes the generic argument of $f$ which is being integrated over.

Case 2: Now suppose that $\mu$ is the counting measure, and $S = \lbrace{1,2,\dots}\rbrace$. Then Holder's inequality becomes:

$$ \sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\frac{1}{p}} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\frac{1}{q}}. $$

Here, the functions $f$ we are integrating have signature $f: \lbrace{1,2,\dots,n\rbrace} \to \mathbb{R}$. Unlike Case 1, the symbol $x = (x_1,x_2,\dots,x_n) \in \mathbb{R}^n$ shows the list of values that result from applying $f$, so that $x_k = f(k)$ for $k \in S$. That is to say we can write the abstract integral that appears in Holder's inequality as:

$$ \lVert{f}\rVert_p = \left(\int_S\mid{f}|^p|d\mu\right)^{1/p} = \left(\sum_{k=1}^{n}|f(k)|^p\right)^{1/p} = \left(\sum_{k=1}^{n}|x_k|^p\right)^{1/p}. $$

Is my interpretation of this situation correct, in particular how the symbol $x$ admits different semantic meaning in Case 1 versus Case 2?

Thinking through this situation seems to suggest that Euclidean space $\mathbb{R}^n$ itself is a function space, where elements $x=(x_1,\dots,x_n)$ of the vector space are in correspondance with functions $f:\lbrace{1,2,\dots,n}\rbrace \to \mathbb{R}$, with $x = (f(1),f(2),\dots,f(n))$. This representation rarely appears explicitly in elementary expositions of Euclidean space.

Is there a pattern in mathematics where many measure theoretic results (other than Holder's inequality) are used to make geometric statements about vectors Euclidean space, assuming the representation of a vector $x$ as function $f$ described above, equipped with the counting measure?


Your interpretation is correct. In the first case, $f$ is the function (with domain $\Bbb R^n$) and $x$ the variable. In the second, $x$ is the function (with domain $\{1,\dots,n\}$) and $k$ the variable. The symbol $x$ is has a different meaning in each case.

As for the last question, I do not think that there is such a pattern. But any abstract measure theoretical result can be applied to $\{1,\dots,n\}$ with the counting measure. Most of the times it will yield a more or less trivial result in that setting.


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