Final

Many exams can be found on the prelims page here

Here are some more practice finals

1. Fall 17
2. Fall 18

Midterm

1. Consider the collection of measurable rectangles in \(\R^2\):

   \(S = \{ (a_1,b_1] \times (a_2,b_2] ~ \colon ~ a_i < b_i ~ i=1,2 \}\)

   Show that $S$ is a semi-ring of sets.

2. Let \(f\) be an integrable function on a measure space. Show that for every \(\epsilon > 0\), there is a \(\delta > 0\) such that for all \(m(A) < \delta\), \(\int_A f < \epsilon\). Hint: Start with \(f \geq 0\), and the identity

   \(\int_A f = \int_A f 1_{f > L} + \int_A f 1_{f \leq L}\)

   Then let $L \to \infty$. Note the “uniformity” in the statement: the integral over \(A\) is small for all small \(m(A)\). This is related to the notion of uniform integrability we will encounter.

3. Let \(f(x) = sin(x)/x\). Show that the (Lebesgue) integral \(\int_{(0,\infty)} f\) is not defined, but \(\lim_{t \to \infty} \int_{(0,t)} f\) exists.

   Hint: First control \(\int_{(0,\pi)} f\) using calculus, and then estimate \(\int_{(2n \pi,(2n + 2)\pi)} f\) using standard inequalities.