exams
Midterm

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 semiring of sets.

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.

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.