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Okay. Give $[0,1]$ Lebesgue measure. For each $t \in [0,1]$ let $S_t = \{1_{\{t\}}\} \cup \{a\cdot 1_{[0,1]}: a \geq 1\}$. Then $\bigcap S_t = \{a\cdot 1_{[0,1]}: a \geq 1\}$ and its inf is the funct …

The question is not clear. Are you asking whether the complete distributive law holds in (probably the unit ball of) $L^\infty(X,\mu)$? The answer is no: complete distributivity is characteristic of a …

This is false. Here's a counterexample. For $n \geq 2$, let $\mu_n$ be the probability measure $\mu_n = (1/n)\delta_0 + (1/2 - 1/n)\delta_1 + (1/2)\delta_n$. The convex hull of $\{\mu_n: n \geq 2\}$ …

Counterexample. First, let ${\cal B}$ be the Borel $\sigma$-algebra on ${\bf R}$ and let ${\cal B}'$ be the $\sigma$-algebra generated by ${\cal B}$ together with one non-Borel set $E$. Note that $E$ …

I gave a simple counterexample in the comments yesterday but I guess it got buried. Let $X$ be the disjoint union of two copies of the long line. The characteristic function of either of them is conti …

It's not a convention, it's a theorem. Let's say I have a measure space $X$ and a function $f: X \to \overline{\mathbb{R}}$ which is identically zero off of a null set $N$, and constantly $+\infty$ on …

Consider the case $\Omega_1 = \Omega_2 = [0,1]$, equipping both factors with Lebesgue measure. Let $S$ be any subset of $[0,1]$. Then the set $$(S \times [0,1/2]) \cup (S^c \times (1/2,1])$$ is "cross …

The answer is yes. Let me explain why. Let $v \in E$ be any nonzero vector. Then define $E_0$ to be the closure of the set of vectors of the form $p(A_1,A_1^*, A_2, A_2^*)v$ where $p$ is a complex po …

If $\mu$ and $\nu$ are, respectively, projection valued measures on $X$ and $Y$, taking values in $B(H)$, then $\mu \otimes \nu$ can be defined to be a projection valued measure on $X\times Y$ taking …

My comment seems to be buried so I'd like to repeat it here. There is a simple C*-algebra construction that answers the question. The quotient space $l^\infty/c_0$ is a unital commutative C*-algebra, …

It sounds like you are new to this subject, so there are some basic things for you to know. The first is that by night all Hilbert spaces are the same, so your $L^2(\Omega)$ can just be a generic $H$. …

An easy way to see this is by using the Lebesgue density theorem. Any set of positive measure has a density point $t$ (indeed, almost every element of the set is a density point). This means that for …

(reading "sequence" as "net", as suggested in the comments) Well, $C_b(\mathbb{R}^+) \cong C(\beta\mathbb{R}^+)$, so any such $L$ will arise from a probability measure on the Stone-Cech remainder $\be …

Partition $(a_n)$ into two subsequences $(a_n')$ and $(a_n'')$ with $\sum a_n' < \infty$, and partition $(d_i)$ into two subsequences $(d_i')$ and $(d_i'')$ such that $d_i'' \to \infty$. Pair the $a_i …

Let $X$ be a metric space consisting of a countable set of points, the distance between any two of which is $2$, together with one additional point $e$ whose distance to any of the other points is $1$ …