This is mostly just a place to store my notes and solutions to Measure, Integral, and Probability, 2nd Edition by Marek Capinski and Peter Kopp.
The overview document will be notes regarding my conceptual approach to measure theory. The specific chapter documents will be my own solutions to the exercises in the book.
Note that there are already solutions available in the back of the book, but they are often only answers and not a full explanation.
Download PDF: Overview Some notes on the taxonomy of spaces and some things to do for me. Topological Spaces Open sets are defined as sets where the $\varepsilon$-ball around each point in the set is contained in the set; i.e., all points are interior points. Closed sets are ones whose complements are open. Equivalently, they are sets that contain all of their limit points. This distinction in sets is useful because it allows to either bound quantities and ensure our set captures some $\varepsilon$-error, or ensure we do not miss any points a sequence converges to, respectively. ...
Download PDF Chapter 2 Exercise 1 Constructing a cover over the null set using only closed intervals: $$ \begin{gather*} \text{Let} \; I_n = [x_n - \frac{\varepsilon}{2^{n+2}}, \:x_n +\frac{\varepsilon}{2^{n+2}}] \implies l(I_n) = \frac{\varepsilon}{2^{n+1}} \\ \text{Then} \; \bigcup_{n = 1}^{\infty}I_n \implies \sum_{n=1}^{\infty}{l\left(I_n\right)} \leq \frac{\varepsilon}{2} < \varepsilon \: \text{(equality when disjoint)}\\ \end{gather*} $$ Since the length of an interval only depends on its endpoints, the above construction works for open and open/closed intervals. Thus, any type of interval is sufficient to cover the null set. ...