The Lyapunov theorem on convexity and its use for sign-embeddings

• Date: 1992
• Main Authors: Kadets , V. М., Popov , М. М.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 1992
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/8169
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

It is proved (Theorem 1) that for a Banach space $X$ the following statements are equialent: i) the range of every $X$-valued $\sigma$-additive non-atomic measure of finite variation has convex closure; ii) $L_1$ does not sign-embed in $X$.