# Proving the Riesz Representation Theorem for the space of 1 point with the counting measure.

I am trying to solve this question:

Give an example of a measure space $$(X, \mathfrak{M}, \mu)$$ for which the Riesz Representation Theorem does extend to the case $$p=\infty.$$

My Trial:

I am trying to mimic the proof in Royden "Real Analysis" fourth edition on page 402.

My example is the space of 1 point i.e. $$X = \{x_{0}\}$$ with $$\mu$$ the counting measure. (taking into account that $$L^{\infty}$$ in this case is the space of all bounded measurable functions $$f: X \rightarrow \mathbb{R}$$ which is the space of constant functions, and that $$L^1$$ in this case is the collection of integrable functions of $$X$$ but we know that $$X$$ is the space of all constant functions which are all integrable.)

Proof:

Since our counting measure equals 1, so we are in the case $$\mu(X)< \infty.$$ Let $$S:L^{\infty}(X, \mu) \rightarrow \mathbb{R}$$ be a bounded linear functional. Define a set function $$\nu$$ on the collection of measurable sets $$\mathfrak{m}$$ by setting $$\nu(E) = S(\chi_{E})$$ for $$E \in \mathfrak{M}.$$this is properly defined since $$\mu(X) < \infty$$ and thus the characteristic function of each measurable set belongs to $$L^{\infty}(X, \mu).$$ We claim that $$\nu$$ is a signed measure.i.e. we need to show that $$\nu(E) = \sum_{k=1}^{\infty} \nu(E_{k})$$ and the series converges absolutely.Indeed, let $$\{E_{k}\}_{k=1}^{\infty}$$ be a countable disjoint collection of measurable sets and $$E = \bigcup_{k=1}^{\infty}E_{k}.$$ By the countable additivity of the measure $$\mu,$$ $$\mu(E) = \sum_{k=1}^{\infty} \mu(E_{k}) < \infty.$$ therefore $$\lim_{n\rightarrow \infty} \sum_{k= n+1}^{\infty} \mu(E_{k}) = 0.$$

Consequently, $$|\nu(E) - \sum_{k=1}^{n}\nu(E_{k})| = |S(\chi_{E}) - \sum_{k=1}^{n}S(\chi_{E_{k}})| = |S(\chi_{E} - \sum_{k=1}^{n} \chi_{E_{k}})|$$

.....

I am just going to mimic the proof in the book.

My question is:

Is there is something special about my example that makes the proof easier (as in the book the proof contains finding the function $$f\in L^1$$as the radon Nikodym derivative of $$\nu$$ with respect to $$\mu$$)

• Yes, this vector space is finite (one) dimensional, and in a finite dimensional vector space any two norms are topologically equivalent. – Berci Feb 17 at 8:26
• So what? how can I complete the proof? @Berci I need to proof the Onto part, I need an isomorphism. what are the amendments that I should make to the given proof in the book? – Mathstupid Feb 17 at 10:04
• It is Riesz representation theorem for the dual of $L^p(X,\mu)$@Berci – Mathstupid Feb 17 at 10:08
• @Berci how about the function $f \in L^1$ that I should construct? – Mathstupid Feb 17 at 10:21
• @Berci Do we have to mimic the proof of the Riesz Representation theorem for the dual of $L^p({x_{0}}, \mu)$ in case of Lebesgue measure instead (on pg.160 of the mentioned book)? – Secretly Feb 17 at 16:20