The completeness in spaces of bounded Pettis integrable functions and in spaces of bounded functions satisfying the law of large numbers (Q2732351)

Let \((\Omega,\Sigma,\mu)\) be a complete probability space, and let \(X\) be a Banach space with the Banach dual space \(X^{\ast}\). For \(f:\Omega\to X\) and for \(x^{\ast}\in X^{\ast}\), denoting the \(L_{\infty}(\mu)\)-norm of \(x^{\ast}f\) by \(\|x^{\ast}f\|_{\infty}\), let \(\|f\|_{\infty}:=\sup\{\|x^{\ast}f\|_{\infty}:x^{\ast}\in X^{\ast}\), \(\|x^{\ast}\|\leq 1\}\). Let \(P_{\infty}(\mu,X)\) be the linear space of weakly equivalent Pettis integrable functions \(f:\Omega\to X\) such that \(\|f\|_{\infty}<\infty\). Assuming that \([0,1]\) cannot be covered by less than the continuum closed sets of the Lebesgue measure zero (which is a consequence of Martin's Axiom), the author proved that \(P_{\infty}(\mu,X^{\ast})\) is complete provided \(\mu\) is perfect. Let \(LLN_{\infty}(\mu,X)\) be the space of weakly equivalent functions \(f:\Omega\to X\) satisfying the law of large numbers and such that \(\|f\|_{\infty}<\infty\). The author also proved that \(LLN_{\infty}(\mu,X^{\ast})\) is always complete.