TMA4225 Foundations of analysis: Fall term 2013

I will on occassion depart from the notation of the book. This page is intended to list the differences.

• I use \(A\subseteq B\) for set containment (\(A\) is a subset of \(B\)) where the book uses \(A\subset B\). To me, the latter indicates strict containment, i.e., \(A\) is contained in \(B\) but the sets are not equal.
• I will use \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\), and \(\mathbb{N}\) for the sets of real numbers, rational numbers, integers, and natural numbers (which start with 1, not 0) respectively. The book uses calligraphic letters \(\mathcal{R}\), \(\mathcal{Q}\), \(\mathcal{Z}\), and \(\mathcal{N}\) for these sets. Oddly, they do use \(\mathbb{C}\) for the complex numbers, as will I.
• If \(\mathcal{C}\) is a set of sets, I just might write \(\bigcup\mathcal{C}\) and \(\bigcup\mathcal{C}\) for the union and intersection of all the members of this set. The book uses the more complicated, but possibly clearer, notation \(\bigcup_{A\in\mathcal{C}}A\) and \(\bigcup_{A\in\mathcal{C}}A\) for these sets. To be clear, \(x\in\bigcup\mathcal{C}\) if and only if \(x\) belongs to some member of \(\mathcal{C}\), while \(x\in\bigcap\mathcal{C}\) if and only if \(x\) belongs to every member of \(\mathcal{C}\).
• I tend to dislike the prefix “non-”, so I avoid it as much as I can. Thus, I prefer “increasing” over “non-decreasing”. I will then say “strictly increasing” if that is what I mean; in particular, a constant function is increasing in my terminology, but not strictly so. Similarly for “decreasing” versus “non-increasing”. I find the negation in the “non-” constructions confusing.
• I write \(\lfloor x\rfloor\) for the greatest integer \(\le x\), and similarly \(\lceil x\rceil\)$ for the smallest integer \(\ge x\). I also call these the floor and ceiling functions, respectively. Note that \(\lceil x\rceil=-\lfloor -x\rfloor\). The book uses \([x]\) for the floor function, which it calls the greatest integer function.
• I prefer to write \(\varlimsup\) and \(\varliminf\) rather than \(\limsup\) and \(\liminf\).
• The concept of neighbourhoods: A neighbourhood of \(a\in\mathbb{R}\) is a subset \(U\subseteq\mathbb{R}^*\) so that for some \(\varepsilon>0\), \(|x-a|<\varepsilon\) implies \(x\in U\). A neighbourhood of \(\infty\) is a subset \(U\subseteq\mathbb{R}^*\) so that for some \(M\in\mathbb{R}\), \(x>M\) implies \(x\in U\). And a neighbourhood of \(-\infty\) is a subset \(U\subseteq\mathbb{R}^*\) so that for some \(M\in\mathbb{R}\), \(x<M\) implies \(x\in U\).
  This concept allows us to unify the treatment of convergence to a real number and convergence to \(\pm\infty\): If \((x_n)_{n\in\mathbb{N}}\) is a sequence in \(\mathbb{R}^*\) and \(a\in\mathbb{R}^*\) then \(x_n\to a\) if and only if for every neighbourhood \(U\) of \(a\), there is some \(N\) so that \(x_n\in U\) whenever \(n\ge N\).
• I sometimes use the symbol \(\bot\) to indicate a contradiction. But I only do this on the blackboard, or in private notes. This is not a universally recognized convention, though it is not uncommon. So it is better avoided in other contexts.
• I use the “square” union sign \(\sqcup\) to denote disjoint union, meaning the union of disjoint sets (or pairwise disjoint, for more than two sets): Thus \(A\sqcup B=A\cup B\), but I reserve this notation for the case where \(A\cap B=\emptyset\) (where the disjointness is either known or assumed). Similarly for \(\bigsqcup_n A_n=\bigcup_n A_n\), which I only use when it is known or assumed that \(A_j\cap A_k=\emptyset\) whenever \(j\ne k\).
• I prefer the notations \(\operatorname{Re}z\) and \(\operatorname{Im}z\) for the real and imaginary parts of a complex number, rather than \(\Re z\) and \(\Im z\).
• I will use the word protomeasure about a function \(\iota\colon\mathscr{C}\to[0,\infty]\) satisfying (E1)–(E3) on p. 180 of the book. And I will call a protomeasure σ-finite if it satisfies (E4) on p. 188 in addition. Protomeasures are exactly those functions that can be extended to a measure, and we have uniqueness of the extension on the generated σ-algebra provided the protomeasure is σ-finite. (I realized later that I called them premeasures in a note. Such things happen when one uses non-standard terminology. Oh well …)
• A note on absolute continuity: The book defines this, in essence as being the indefinite integral of some \(\mathscr{L}^1\) function. I will instead follow convention, and define absolute continuity via the \(\epsilon\)-\(\delta\) property seen in Proposition 8.6 on p. 294. Thus the statement that a function is an indefinite integral if and only if it is absolutely continuous becomes a theorem (the content of propositions 8.6 and 8.7) and not a definition.
• Further differences will be added here as I notice them.

(This section is written in Norwegian, for obvious reasons.)

Her har jeg tenkt å samle noen oversettelser til norsk av fagtermer i kurset. Listen vil neppe være komplett noen gang. Spør meg om det er noe du ikke kjenner et norsk ord for, så skal jeg forsøke å legge det til. Jeg tar ikke med termer med åenbare oversettelser, som monotone class (monoton klasse) – hvis det ikke blir åpenbart at mange er i tvil om det rette.