Takehome Midterm

Due Date: Sunday, April 3 at 11:59 pm

Rules: This is a midterm exam, not a homework assignment. You must solve the problems entirely on your own, and you should not discuss the problems with any other students in the class. While working on the exam, you should feel free to consult the notes and homework solutions posted on the class web page, as well as any notes you might have related to the class. If you like, you may also consult the following textbooks:

• The Real Numbers and Real Analysis by Ethan Bloch.
• Proofs & Fundamentals: A First Course in Abstract Mathematics by Ethan Bloch.
• Calculus by James Stewart.
• Topology by James Munkres.
• Principles of Mathematical Analysis by Walter Rudin.
• Real & Complex Analysis by Walter Rudin
• A Primer of Lebesgue Integration by H. S. Bear.
• An Introduction to Measure Theory by Terence Tao.

You should not consult any other textbooks or internet sources when working on the exam.

1. If \(S\) and \(T\) are sets, the symmetric difference of \(S\) and \(T\) is the set \[ S \bigtriangleup T \,=\, (S\cap T^c) \cup (S^c \cap T). \] If \(\{E_n\}\) is a sequence of measurable sets in \([0,1]\), we say that \(\{E_n\}\) converges to a measurable set \(E\) if \[ \lim_{n\to\infty} m(E_n \bigtriangleup E) \,=\, 0. \]
   1. Prove that if \(E_1,E_2,\ldots\) and \(E\) are measurable sets in \([0,1]\) and \(\chi_{E_n} \to \chi_E\) pointwise almost everywhere on \([0,1]\), then \(\{E_n\}\) converges to \(E\).
   2. Find a sequence \(\{E_n\}\) of measurable sets in \([0,1]\) such that \(\{E_n\}\) converges to the empty set but \(\{\chi_{E_n}(x)\}\) does not converge for any \(x\in[0,1]\).
   3. Prove that if \(\{E_n\}\) is a sequence of measurable sets in \([0,1]\) and \[ \sum_{n=1}^\infty m(E_n\bigtriangleup E_{n+1}) \,<\, \infty, \] then \(\{\chi_{E_n}\}\) converges pointwise almost everywhere.
   4. Let \(\{E_n\}\) be a sequence of measurable sets in \([0,1]\). Suppose that for every \(\epsilon > 0\), there exists an \(N\in\mathbb{N}\) so that \[ i,j \geq N\qquad\Rightarrow\qquad m(E_i \bigtriangleup E_j) < \epsilon. \] Use parts (a) and (c) to prove that \(\{E_n\}\) converges to some measurable set \(E\).
2. If \(f\colon\mathbb{R}\to\mathbb{R}\) is a bounded measurable function and \(g\colon\mathbb{R}\to\mathbb{R}\) is an \(L^1\) function, the convolution of \(f\) and \(g\) is the function \(f*g\colon \mathbb{R}\to\mathbb{R}\) defined by \[ (f*g)(x) \,=\, \int_{\mathbb{R}} f(x-t)\,g(t)\,dm(t). \] (Note that this integral always exists, so \(f*g\) is a well-defined function.)
   1. Let \(f_n\colon \mathbb{R}\to\mathbb{R}\) be a uniformly bounded sequence of measurable functions converging pointwise to a function \(f\colon\mathbb{R}\to\mathbb{R}\), and let \(g\colon\mathbb{R}\to\mathbb{R}\) be an \(L^1\) function. Prove that \(f_n*g\) converges pointwise to \(f*g\).
   2. Prove that if \(f\colon\mathbb{R}\to\mathbb{R}\) is a bounded continuous function and \(g\colon \mathbb{R}\to\mathbb{R}\) is an \(L^1\) function, then \(f*g\) is continuous.