An -valued function’s intermediate value theorem and its applications to random uniform convexity

• Published in: Acta mathematica Sinica. English series Vol. 28; no. 5; pp. 909 - 924
• Main Authors: Guo, Tie Xin, Zeng, Xiao Lin
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 2012
• Subjects: Mathematics; Mathematics and Statistics; random normed module; ℝ)-valued function; modulus of random convexity; 46E30; (; 46B20; random uniform convexity; 46A22; intermediate value theorem
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-011-0367-2

Let be a probability space and the algebra of equivalence classes of realvalued random variables on . When is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from to . As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S , ‖·‖) is random uniformly convex iff L p ( S ) is uniformly convex for each fixed positive number p such that 1 < p < +∞.