I am trying to understand the following lemma from Royden's Real Analysis which is directly used to prove Riesz Representation theorem.
The book in the proof states that " when $p = 1$. We must show that $M$ is an essential upper bound for $f"$. Can anyone explain why we must show that $M$ in ($9$) is an essential upper bound? If $p$ is not one, is this still true?
Please bear with me if this is too basic. For some reason, i just cannot get my head around this.
When $p=1$ the conjugate exponent is $q=\infty$, as $\frac{1}{1}+\frac{1}{\infty}=1$ for this purpose. To say $f\in L^\infty$ means $f$ is essentially bounded, ie. there is an $M$ such that $|f(x)|\leq M$ $\mu$-almost everywhere. This is just a definition so perhaps you're overthinking it.