# Study Guide

References:

• Folland’s “Real Analysis: Modern Techniques”, Ch.1
• Stein and Shakarchi Ch.1, Ch.2

## Convergence Tips/Tricks

• Our favorite tools: metrics and norms!
• So show things are equal by showing $${\left\lvert {x-y} \right\rvert} = 0$$. Know the triangle inequality by heart!
• Uniform convergence:
• Negating: find a bad $${\varepsilon}$$ and a single bad point $$x$$.
• Showing a sum converges uniformly: remember that $$\sum_{k\geq 1} a_k$$ is defined to be $$\lim_{N\to\infty} \sum_{k\leq N} a_k$$. So the trick is to define $$f_n(x) := \sum_{k\leq n} a_k$$ and then apply the usual criteria above.
• It’s sometimes useful to trade the $$\forall x$$ in the definition with $$\sup_{x\in X} {\left\lvert {f_n(x) - f(x)} \right\rvert} < {\varepsilon}$$ instead.
• Compare and contrast to pointwise convergence, which is strictly weaker:
• The main difference: pointwise can depend on the $$x$$ and the $${\varepsilon}$$, but uniform needs one $${\varepsilon}$$ that works for all $$x$$ simultaneously.
• Note uniform implies pointwise but not conversely.
• The sup norm: $${\left\lVert {f} \right\rVert}_\infty := \sup_{x\in X} {\left\lvert {f_n(x)} \right\rvert}$$
• A useful way to force uniform convergence: bound your sequence uniformly by a sequence that goes to zero:
• Sups and infs: sup is the least upper bound, inf is the greatest lower bound.
• The $$p-$$test: \begin{align*} \sum_{n\geq 1} {1 \over n^p} < \infty \iff p>1 \end{align*}
• Useful fact: convergent sums have small tails, i.e.  \begin{align*} \sum_{n\geq 1} a_n < \infty \implies \lim_{N\to\infty}\sum_{n\geq N} a_n = 0 \end{align*}
• So try bounding things from above by the tail of a sum!
• If you can’t bound by a tail: as long as you have control over the coefficients, you can pick them to make the sum to converge “fast enough”.
• Example: for a fixed $${\varepsilon}$$, choose $$a_n = 1/2^n$$. Note that $$\sum_{n\geq 1} 1/2^n = 1$$, so choose $$a_n := {\varepsilon}/2^n$$: \begin{align*} \cdots \leq \sum_{n\geq 1} a_n := \sum_{n\geq 1} {{\varepsilon}\over 2^n} = {\varepsilon}\to 0 \end{align*}
• The $${\varepsilon}/3$$ trick:
• The $$M{\hbox{-}}$$test:

## Measure Theory

• $$F_\sigma$$ sets: unions of closed sets ($$F$$ for fermi, French for closed. Sigma for sums, ie unions)
• $$G_\delta$$ sets: intersections of open sets
• $$\sigma$$ algebras: closed under complements, countable intersections, countable unions
• Some of the most useful properties of measures:

• The proof of continuity of measure contains a very useful trick: replace a sequence of sets $$\left\{{E_k}\right\}$$ with a sequence of disjoint sets that either union or intersect to the same thing.
• Example: if $$A_1 \subseteq A_2 \subseteq \cdots$$, set $$F_1=A_1$$ and $$F_k = A_k \setminus A_{k-1}$$ for $$k\geq 2$$. Then $$\bigcup_{k\geq 1} A_k = \coprod_{k\geq 1} F_k$$.
• Occasionally you need some properties of outer measures:

• Outer measure for $${\mathbf{R}}^n$$: you consider all collections of cubes that cover your set, sum up their volumes, and take the infimum over all such collections:

• “Almost everywhere blah” : the set where blah does not happen has measure zero.
• “Infinitely many/all but finitely many” types of sets, which show up in Borel-Cantelli style problems

• Lemmas that sometimes show up on quals:

## Fubini-Tonelli

Quick statement:

Explained in Stein and Shakarchi (Fubini, which requires integrability)

And Tonelli, which only requires measurability:

A more precise statement from Folland:

Some things that qual questions are commonly based on:

# Qual Problems

Suggested by Peter Woolfitt!

Spring 2012

Fall 2016.2

Fall 2018.5

Spring 2019.4: This is an expanded version of Fall 2018 #5 above.