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I give an elementary solution to Problem 1 in $\mathbb{R}^n$ in my book Measure Theory and Functional Analysis (Proposition 2.16, p. 48). Here is the one-dimensional version. I guess it's clear that t …

Edit: the proof can be made a little simpler. Yes, this condition is equivalent to $f$ being continuous. The reverse direction is easy because if $f$ is continuous at $x$ then all of the limits in que …

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 …

Counterexample. For each $n$ let $k_n$ be the characteristic function of $[0,\frac{1}{2^n}] \cup [\frac{2}{2^n},\frac{3}{2^n}] \cup \cdots$. Next observe that there are only countably many subsets of …

It helps to note that $\mathcal{T}_tf(s) = K(s,t)\langle f(\cdot), \overline{K}(\cdot,t)\rangle$. If $K_n = \sum a_i1_{A_i\times B_i}$ is a finite linear combination of characteristic functions of rec …

There exists a continuous, bounded utility function if and only if the relation is continuous in the stronger sense of being closed in $P(X) \times P(X)$, using the weak${}^*$ topology on each factor. …

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 $1$ …

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 …

(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 …

Assuming $C$ isn't also open, find a sequence $(x_n)$ in $X\setminus C$ which converges to a point $x$ in $C$. Then the point measures $\delta_{x_n}$ converge to $\delta_x$ but their integrals against …

It isn't research level, but $f(t) = \sin(e^{t^2})$ is a counterexample. (The uniform distance between $f$ and any shift of $f$ is $1$.)

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 $f_n \to f$ …

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 …

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 …