Ph.D. Preliminary Examination in Real Analysis

Let \((X, \mathcal {M}, \mu )\) be a measure space and \(f : X \to Y\) be a function.

1. Show that \(\mathcal {N} = \{f^{-1}A : A \in \mathcal {M}\}\) is a \(\sigma \)-algebra.
2. Let \(\nu (B) = \mu (f(B))\) for \(B \in \mathcal {N}\). Show that \(\nu \) is a measure.
3. Show that even if \((X, \mathcal {M}, \mu )\) is complete we do not necessarily have that \((Y, \mathcal {N}, \nu )\) is complete.

Define the weak topology on \(L^{2}(\mathbb {R}, \lambda )\) and prove that it is not metrizable.

Let \(f : [0, 1] \to [0, \infty ]\) be measurable. Show that \(U(f) = \{(x, y) : 0 \le y \le f(x)\}\) is measurable and \[ \int f\,d\lambda = \lambda ^{2}(U(f)). \]

State the Lebesgue dominated convergence theorem. Show that the converse is not true. That is, there is a sequence of functions that satisfy the conclusion of the Lebesgue dominated convergence theorem but not its assumption.

Let \((X, \mathcal {T})\) be a topological space and \(\mathcal {B}\) be its Borel \(\sigma \)-algebra. Show that if \((X, \mathcal {B}, \mu )\) is a \(\sigma \)-finite measure space that is outer regular on finite measure, measurable subsets, then it is automatically outer regular on all measurable subsets. Recall that \(\mu \) is called outer regular on a collection of subsets \(\mathcal {S}\) if for every \(A \in \mathcal {S}\) and \(\epsilon > 0\) there is an open set \(U\) so that \(A \subset U\) and \(\mu (U \setminus A) < \epsilon \).

State the open mapping theorem for Banach spaces and use it to prove the following: Let \((B, \|\cdot \|_{B})\) and \((C, \|\cdot \|_{C})\) be Banach spaces and \(A : B \to C\) be continuous and linear. If \(A\) is a bijection then \(A^{-1}\) is continuous.