Question in Proof of Riesz Representation Theorem

Question

I'm trying to give a step-by-step proof of the Riesz Representation Theorem for the Dual of $L^{1}(X,\mu)$ and I've hit a wall trying to show $T:L^{\infty}(X,\mu)\to (L^{1}(X,\mu))^{*}$ defined by $T(f)=T_{f}$ where $$T_{f}(g)=\int_{X}f\cdot g\,d\mu, \quad \forall g\in L^{1}(X,\mu)$$ is an isometry. The book is saying that I should have $\|T_{f}\|=\|f\|_{\infty}$ (which supposedly follows from Hölder's Inequality). I've shown that $T$ is bounded, and in doing so I've already shown that $\| T_{f} \| \leq \|f\|_{\infty}.$ However I'm not sure how to show the reverse inequality. I've tried using the definitions of the operator norm and of the essential supremum, to no avail. Could someone please point me in the right direction? (Note: I'm using Royden & Fitzpatrick's Real Analysis, 4 ed.)

Answer 1

Hint Let $$A:= \{ x \in X | |f(x)| \geq \| f\ |_\infty -\epsilon\} \,.$$ Then $A$ has positive measure.

Define $g(x)=\frac{1}{\mu(A)} 1_A(x) \mbox{sign}(f(x)) $