Exercise 8.1.6

Question

Consider the interval $[0,1]$.
\begin{enumerate}[label=(\alph*)] 
\item If $\delta(x) = 1/9$, find a $\delta(x)$-fine tagged partition of $[0,1]$. Does the choice of tags matter in this case?
\item Let
$$\delta(x) = \begin{cases}
   1/4 & \text{if } x = 0 \
   x/3 & \text{if } 0 < x \leq 1
\end{cases}$$
Construct a $\delta(x)$-fine tagged partition of $[0,1]$.
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item $P = \{n/10 : n \in \mathbf{N}, 0 \leq n \leq 10\}$. Choice of tags does not matter since $\delta(x)$ is constant.
\item $P = \{n/15 : n \in \mathbf{N}, 3 \leq n \leq 15\} \cup \{0\}$, $c_1 = 0$, $c_k = (k+2)/15$ for $k \geq 2$. Then $\delta(c_k) > 1/15$ for $k \geq 2$.
 \end{enumerate}

Exercise 8.1.6

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Exercise 8.1.6

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