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It's notable that most of the "bread and butter" mathematical consequences of the axiom of choice are actually consequences of countable choice. (Every infinite set contains a countable subset, a coun …

If $f: \mathbb{R} \to [0,\infty)$ is Borel (or Lebesgue) measurable, then for each rational $a > 0$ define $X_a = f^{-1}([a,\infty)) \times [0,a)$. Then each $X_a$ is measurable and their union is exa …

Let $A$ be a measurable subset of $[0,1]$ such that both it and its complement have positive measure in every open interval in $[0,1]$ (see here for example). Its characteristic function is dominated …

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, …

I propose the axiom that any isometry between two such sets must take the center of one onto the center of the other. This axiom by itself is consistent with the existence of a center for every weakly …

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 …

This isn't a direct answer, but it may be on topic, depending on the motivation for the question. When I teach measure theory, I feel I owe the students an explanation of why they should have to lear …

This is known as the Arens-Eells space $AE(X)$. In the nonlinear Banach space literature it's also called the Lipschitz-free space $\mathcal{F}(X)$. It is not a dual space in general, but rather the p …

No. The bounded Baire class one functions on $[0,1]$ are stable under uniform limits and hence constitute a C*-algebra. This C*-algebra contains every semicontinuous function on [0,1]. Every function …

Yes. If $f_n \to f$ weak* then the sequence $(f_n)$ must be bounded in ${\rm Lip}_0(X)$ (Banach-Steinhaus), and for bounded nets weak* convergence is the same as pointwise convergence. So $f_n \to f$ …

(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 …

For the sake of readability, I am going to make this a separate answer. In response to my other answer, Bogdan points out that preservation under isometries need not determine the center. I suppose in …

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 …

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 $1$ …

This fails for $X = \mathbb{R}$, and hence for every nonzero Banach space, since they all contain copies of $\mathbb{R}$. If the map $t \mapsto \delta_t$ were differentiable in either sense then for e …