User rhymer - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:43:47Z http://mathoverflow.net/feeds/user/15719 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67511/characterization-of-weakly-measurable-functions Characterization of Weakly measurable functions Rhymer 2011-06-11T14:14:52Z 2011-06-12T08:31:09Z <p>I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong measurability? </p> <p>More specifically, a function $f:\Omega\mapsto X$ defined on a measure space $(\Omega,\Sigma,\mu)$ and taking values in a Banach space $X$ is <em>strongly measurable</em> if there exists a sequence of step functions $\{ \phi_n \}$ such that $\phi_n\rightarrow f$ in norm a.e.. Could we analogously say that $f$ is <em>weakly measurable</em> iff there exists a sequence of step functions $\{ \phi_n \}$ such that $\phi_n\rightarrow f$ weakly a.e.? One direction is obviously true, but I can't figure out the other direction.</p> <p>For reference, here is the definition of weak measurability: A function $f:\Omega\mapsto X$ is <em>weakly measurable</em> if $\langle f(\omega), x \rangle$ is measurable for each $x\in X'$, the norm dual of $X$.</p> http://mathoverflow.net/questions/67511/characterization-of-weakly-measurable-functions/67514#67514 Comment by Rhymer Rhymer 2011-06-13T06:45:32Z 2011-06-13T06:45:32Z @ Michael Renardy: Thank you for the great explanation! @ TaQ: Thank you for your comments, I was about to ask the same question!