Characterization of Weakly measurable functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:17:22Z http://mathoverflow.net/feeds/question/67511 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 Answer by Michael Renardy for Characterization of Weakly measurable functions Michael Renardy 2011-06-11T14:38:28Z 2011-06-11T14:38:28Z <p>If there is a sequence of step functions such that $\phi_n\to f$ weakly a.e., then $f$ is almost separably valued. But if it is weakly measurable and almost separably valued, it is strongly measurable.</p> http://mathoverflow.net/questions/67511/characterization-of-weakly-measurable-functions/67520#67520 Answer by Gerald Edgar for Characterization of Weakly measurable functions Gerald Edgar 2011-06-11T16:18:10Z 2011-06-11T16:32:35Z <p><strong>Example 1:</strong> $f : [0,1] \to l^2[0,1]$ that is not almost separably valued: $f(x) = \delta_x$, the function equal to $1$ at $x$ and zero elsewhere. At least this one is <em>scalarly equivalent</em> to the constant zero. </p> <p><strong>Example 2:</strong> (page 672 of [1] where details are found) $f : [0,1] \to L^\infty[0,1]$ with $f(x) = 1_{[0,x]}$, the characteristic function of $[0,x]$. Then $f$ is scalarly measurable but not scalarly equivalent to a Bochner measurable function.</p> <p>my references on measurability in Banach space: </p> <p>[1] Indiana Univ Math J. 26 (1977) 663--667</p> <p>[2] Indiana Univ Math J. 28 (1979) 559--579 </p> <p>Here I have used the terms "scalarly measurable" and "Bochner measurable" in place of weakly and strongly measurable.</p>