Kahane’s construction and the weak sequential completeness of $L^{1}$

Abstract:

If $\lambda$ is a normalized Lebesgue measure $(\text {arc length}/2\pi )$ on $T = \{ |z| = 1\}$, the Gelfand map permits ${L^\infty }{(\lambda )^ \ast }$ to be identified with $M(X)$, the space of finite Baire measures on $X$, the maximal ideal space of ${L^\infty }(\lambda )$. The measure ${m_0}$ in $M(X)$ represents $\lambda :\int {X\hat fd{m_0} = \int {Tfd\lambda } }$ for all $f \in {L^\infty }(\lambda )$. Furthermore $\mu \ll {m_0}$ if and only if $\mu$ represents some measure of the form $\phi d\lambda ,\phi \in {L^1}(\lambda )$. Using this fact and a sum constructed by J. P. Kahane, $\Sigma {\hat h^{n(j)}}$ when $\hat h$ is an appropriate function guaranteed by Urysohn’s lemma, develops a proof that ${L^1}(\lambda )$ is weakly sequentially complete.