I'm trying to solve this question:

Give an example of a measuring room $ (X, mathfrak {M}, mu) $ for which the Riesz representation theorem extends to the case $ p = infty. $

**My process:**

I'm trying to mimic the evidence in Royden's fourth edition of "Real Analysis" on page 402.

My example is the space of 1 point, i.e. $ X = {x_ {0} } $ With $ mu $ the count. (taking that into account $ L ^ { infty} $ in this case is the space of all limited measurable functions $ f: X rightarrow mathbb {R} $ Which is the space of constant functions, and that $ L ^ 1 $ in this case the collection of integrable functions from $ X $ but we know that $ X $ is the space of all constant functions, all of which can be integrated.)

**Proof:**

Since our count is 1, we are in the case $ mu (X) < infty. $ To let $ S: L ^ { infty} (X, mu) rightarrow mathbb {R} $ be a limited linear function. Define a set function $ nu $ on the collection of measurable quantities $ mathfrak {m} $ by setting it $$ nu (E) = S ( chi_ {E}) $$ to the $ E in mathfrak {M}. $that is correctly defined $ mu (X) < infty $ and thus the characteristic function of every measurable quantity is part of it $ L ^ { infty} (X, mu). $ We claim that $ nu $ is a signed measure. we have to show that $$ nu (E) = sum_ {k = 1} ^ { infty} nu (E_ {k}) $$ and the series absolutely converges $ {E_ {k} } _ {k = 1} ^ { infty} $ be a countable disjoint collection of measurable quantities and $ E = bigcup_ {k = 1} ^ { infty} E_ {k}. $ Through 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'm just going to mimic the evidence in the book.

**My question is:**

Is there anything special about my example that makes the proof easier (like in the book does the proof include finding the function? $ f in L ^ 1 $as a radon-nicodymium derivative of $ nu $ in memory of $ mu $)