Explaining maths

This post is also available in PDF form from my WordPress Blog at  https://explainingmaths.wordpress.com/2023/09/01/revisiting-liminf-and-limsup/ My recent series of posts about Fatou's Lemma for sums was aimed primarily at mathematics students at 3rd-year undergraduate level or above. So I think I should write a post more suitable for first-year undergraduate students. So I'm going to have another look at the topic of $\liminf$ and $\limsup$ for bounded sequences of real numbers (though most of what I say can be generalised to sequences, or even nets, of extended real numbers). I said quite a lot about this in a previous post on my WordPress blog in the post https://explainingmaths.wordpress.com/2009/01/10/an-application-of-absorption-to-teaching-lim-inf-and-lim-sup/ and there was a lot of discussion in the comments below the post. In particular, in those comments, I discussed the $\liminf$ and the $\limsup$ of some examples of sequences $(x_n)$, including the following exam...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I will make a PDF of the main article available via my WordPress blog In this appendix, following on from Appendices 3 and 4, we see how the argument from Part IV of the main series gives us Fatou's Lemma for sums without using Measure and Integration. As in Appendix 3, we will assume that we know standard facts about finite sums. We will use the same definitions as in Part IV concerning the $\liminf$ of nets in $\Rbar$ and of nets of functions from $X$ to $\Rbar$. We base our arguments on the following facts about sums (see earlier posts). Let $X$ be a set, and let $f$ be a function from $X$ to $[0,\infty]$. Then \[\sum_{x \in X} f(x) = \sup \left\{ \sum_{x \in E} f(x): E \textrm{ is a finite subset of } X \right\}.\tag{$1$}\] It follows that, whenever $B_1$ and $B_2$ are su...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I will make a PDF of the main article available via my  WordPress blog In this appendix, we fill in some details about claims made in Part IV concerning $\liminf$. We will use the same definitions as in Part IV concerning the $\liminf$ of nets in $\Rbar$. Now let $J$ be a (non-empty) indexing set, and suppose that we have non-negative extended real numbers $(a_j)_{j \in J}$ and $(b_j)_{j \in J}$. We claim that \[\inf\{a_j + b_j: j \in J\}\geq \inf\{a_j:j \in J\} + \inf\{b_j:j\in J\}.\tag{$1$}\] This is fairly standard, but here is a quick proof for those in any doubt. Set $a=\inf\{a_j:j \in J\}$ and $b=\inf\{b_j:j\in J\}$. Then, for each $j \in J$, we have $a_j \geq a$ and $b_j \geq b$. Thus (noting that all values are in $[0,\infty]$, so there is no danger when adding) we ha...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I will make a PDF of the main article available via my  WordPress blog In this appendix, I'll look at defining sums over general sets without referring to Measure and Integration. Our starting point will be the following. Let $X$ be a set and let $f$ be a function from $X$ to $[0,\infty]$. Although we are using non-negative extended real numbers, nevertheless the arguments here will have more of a first-year flavour than some of the other posts in this particular series, while continuing with the theme of bootstrapping from results for finite sets to results for general sets. If $X$ is a finite set, we define $\sum_{x \in X} f(x)$ in the usual way, and we take it as standard that we obtain the same value whichever order we add the values. (By convention, the empty sum is $0$....

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I will make a PDF of the main article available via my  WordPress blog Let $\mu$ be counting measure on (the power set of) a set $X$, and let $f$ be a function from $X$ to $[0,\infty]$. In Part I we defined  \[\displaystyle \sum_{x \in X} f(x) = \int_X f\,\mathrm{d}\mu\,,\] and we noted that this agreed with our usual notions if the set $X$ is countable. We then claimed that (even if $X$ is uncountable), we have \[\sum_{x \in X} f(x) = \sup \left\{ \sum_{x \in E} f(x): E \textrm{ is a finite subset of } X \right\}.\tag{$1$}\] The aim of this post is to prove ($1$). In terms of integrals, we can rewrite $(1)$ as \[\int_X f\,\rd\mu= \sup \left\{\int_E f\,\rd\mu: E \textrm{ is a finite subset of } X \right\}\,.\tag{$2$}\] This is the form we used in Part IV of this series. ...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I will make a PDF of the main article available via my  WordPress blog Counting measure is a very standard example of a measure, but not everyone checks the details. I'm going to attempt to give a fairly efficient proof! But here are a few comments first about what we will be assuming before we start. By convention the empty set, $\emptyset$, is a finite set, and the number of points in $\emptyset$ is $0$. For a finite set $A$, we denote the cardinality of $A$ (number of points in $A$) by $|A|$. Here $|A|$ is a non-negative integer, and $|A|=0$ if and only if $A=\emptyset$. A finite union of finite sets is finite. If we have finitely many pairwise disjoint  finite sets $A_1, A_2, \dots, A_m$, then \[\left|\bigcup_{k=1}^m A_k \right| = \sum_{k=1}^m |A_k|\,.\] It is, of co...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I have made a PDF of the main article available via my  WordPress blog This post is a little more advanced, in as much as it assumes some knowledge of nets rather than just sequences. In this post we will look at the three big convergence theorems for Lebesgue integration, concerning sequences of functions, and see how these go wrong if you use nets instead. Then we will see that they go right again (for nets) if we work with the special case of counting measure. Let's jump straight in with an example of a 'bad' net of functions when working with Lebesgue measure $\lambda$ on $[0,1]$. Our directed set will be the set $\mathcal{F}$ of all finite subsets of $[0,1]$, partially ordered by inclusion. Our net of functions $g_E\jq(E\in\mathcal{F})$ will just be the character...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I have made a PDF of the main article available via my  WordPress blog This post follows on from Part IIIa , where we introduced the notation and explained the setting we are working in. In particular, we are working with sets $\N=\{1,2,3,\dots\}$, $\N_0=\N\cup\{0\}=\{0,1,2,\dots\}$, $X=\N_0 \times \N_0$, and with an array of non-negative extended real numbers $a_{m,n} \jq(m,n \in \mathbb{N_0})$. Our aim is to prove that  \[\sum_{m=0}^\infty \sum_{n=0}^\infty a_{m,n} = \sum_{n=0}^\infty \sum_{m=0}^\infty a_{m,n} = \sum_{k=0}^\infty \sum_{j=0}^{k} a_{j,k-j}\,.\tag{1}\] We take $\nu$ to be counting measure on $X=\N_0 \times \N_0$, and define $F:X\rightarrow [0,\infty]$ by $F(m,n)=a_{m,n}\jq(m,n \in \mathbb{N_0})\,.$ We plan to prove that all of the sums in $(1)$ are equal ...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure If the MathJax mathematics below does not display properly in your browser, a PDF of the main article is available via my WordPress blog In the last post we saw that, when we sum a series of non-negative extended real numbers, whichever order we choose to add the terms in we still obtain the same result, namely the integral of the relevant function against (or with respect to) counting measure. In this post we will look at adding up an infinite  array  of non-negative extended real numbers.  Recall that, for us, $\N=\{1,2,3,\dots\}$. For convenience we denote by $\N_0$ the set of non-negative integers, so that  $\N_0=\N\cup\{0\}=\{0,1,2,\dots\}$. Below we will want to use counting measure on various sets simultaneously, so we use a variety of symbols. However the sigma algebras we use will always be the whole of the power set of each set in...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I have made a PDF of the main article available via my  WordPress blog  at  https://explainingmaths.files.wordpress.com/2023/08/sums-and-integration-against-counting-measure_-part-ii.pdf In this post we will recover some familiar results about summing non-negative real numbers. In fact we will work with non-negative extended real numbers, and see how to interpret the usual results in terms of integration against counting measure. Let $X$ be a countably infinite set (for example $\mathbb{N}$), let $\mu$ be counting measure on $X$ (or more formally on $\mathcal{P}(X)$), and let $f:X\rightarrow [0,\infty].$ In our previous post, we said that we could define \[\sum_{x \in X} f(x) = \int_X f\,\mathrm{d}\mu\,.\] An alternative is to consider a bijection $g:\mathbb{N}\to X...

For other posts in this series, see https://explaining-maths.blogspot.com/search?q=Sums+and+integration+against+counting+measure In case the MathJax mathematics below does not display properly in your browser, I have made a PDF of the main article available via my  WordPress blog  at  https://explainingmaths.files.wordpress.com/2023/08/sums-and-integration-against-counting-measure_-part-i.pdf Note added 30/8/23: I'm not quite sure when I started to say "integration against a measure". I'm not alone, but I think that "integration with respect to a measure" is a lot more common! However, I don't want to change the titles of my latest 10 blog posts, so I'll leave it as it is. I'll probably change back to "integration   with respect to a measure" in future though! It has been a while since I wrote a mathematical post! Here I thought I would say something about Measure and Integration which is relevant to my research. For these posts I as...