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Author: lenis2000

graduate/exams/analysis/2007Aug.html

@@ -200,32 +200,32 @@ <h1 class="unnumbered" id="real-and-complex-analysis-general-exam-fall-2007">Rea
 
 <h2 class="unnumbered" id="problem-1">Problem 1</h2>
 <p>Suppose that (
-<math role="math" aria-label="X, \mathcal{A}, \mu" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>,</mo><mi>𝒜</mi><mo>,</mo><mi>&mu;</mi></mrow><annotation encoding="application/x-tex">X, \mathcal{A}, \mu</annotation></semantics></math>
+<math role="math" aria-label="X, \mathcal{A}, \mu" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>,</mo><mi>&Ascr;</mi><mo>,</mo><mi>&mu;</mi></mrow><annotation encoding="application/x-tex">X, \mathcal{A}, \mu</annotation></semantics></math>
 ) is a measure space, with
-<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝒜</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>
+<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&Ascr;</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>
 a
 <math role="math" aria-label="\sigma" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&sigma;</mi><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra,
 <math role="math" aria-label="\mu" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&mu;</mi><annotation encoding="application/x-tex">\mu</annotation></semantics></math>
 a finite measure. Let
-<math role="math" aria-label="\mathcal{A}_{0} \subset \mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>𝒜</mi><mn>0</mn></msub><mo>&sub;</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}_{0} \subset \mathcal{A}</annotation></semantics></math>
+<math role="math" aria-label="\mathcal{A}_{0} \subset \mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>&Ascr;</mi><mn>0</mn></msub><mo>&sub;</mo><mi>&Ascr;</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}_{0} \subset \mathcal{A}</annotation></semantics></math>
 be a sub-algebra of sets which generates
-<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝒜</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>;
+<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&Ascr;</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>;
 thus
-<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>𝒜</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>
+<math role="math" aria-label="\mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&Ascr;</mi><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>
 is the smallest
 <math role="math" aria-label="\sigma" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&sigma;</mi><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra
 generated by
-<math role="math" aria-label="\mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>𝒜</mi><mn>0</mn></msub><annotation encoding="application/x-tex">\mathcal{A}_{0}</annotation></semantics></math>.<br />
+<math role="math" aria-label="\mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>&Ascr;</mi><mn>0</mn></msub><annotation encoding="application/x-tex">\mathcal{A}_{0}</annotation></semantics></math>.<br />
 Let
-<math role="math" aria-label="\mathcal{B} \subset \mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ℬ</mi><mo>&sub;</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} \subset \mathcal{A}</annotation></semantics></math>
+<math role="math" aria-label="\mathcal{B} \subset \mathcal{A}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ℬ</mi><mo>&sub;</mo><mi>&Ascr;</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} \subset \mathcal{A}</annotation></semantics></math>
 be the collection of subsets of
 <math role="math" aria-label="X" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math>
 with the property that for every
 <math role="math" aria-label="\epsilon" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>ϵ</mi><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>
 and
 <math role="math" aria-label="B \in \mathcal{B}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>&isin;</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">B \in \mathcal{B}</annotation></semantics></math>,
 there is an
-<math role="math" aria-label="A \in \mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&isin;</mo><msub><mi>𝒜</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}_{0}</annotation></semantics></math>,
+<math role="math" aria-label="A \in \mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&isin;</mo><msub><mi>&Ascr;</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}_{0}</annotation></semantics></math>,
 with</p>
 <p><math role="math" aria-label="\mu(A \Delta B)&lt;\epsilon ." display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&mu;</mi><mo stretchy="false" form="prefix">(</mo><mi>A</mi><mi mathvariant="normal">&Delta;</mi><mi>B</mi><mo stretchy="false" form="postfix">)</mo><mo>&lt;</mo><mi>ϵ</mi><mi>.</mi></mrow><annotation encoding="application/x-tex">\mu(A \Delta B)&lt;\epsilon .</annotation></semantics></math></p>
 <p>(Here,
@@ -237,7 +237,7 @@ <h2 class="unnumbered" id="problem-1">Problem 1</h2>
 is a
 <math role="math" aria-label="\sigma" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&sigma;</mi><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-algebra
 containing
-<math role="math" aria-label="\mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>𝒜</mi><mn>0</mn></msub><annotation encoding="application/x-tex">\mathcal{A}_{0}</annotation></semantics></math>,
+<math role="math" aria-label="\mathcal{A}_{0}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>&Ascr;</mi><mn>0</mn></msub><annotation encoding="application/x-tex">\mathcal{A}_{0}</annotation></semantics></math>,
 in particular that it is an algebra, and that it is closed under
 countable (increasing) unions.</p>
 <p>Hint: You may assume that (do not prove!):</p>
@@ -355,7 +355,7 @@ <h2 class="unnumbered" id="problem-5">Problem 5</h2>
 be the Hilbert space of
 <math role="math" aria-label="L^{2}(D, \mu)" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mn>2</mn></msup><mo stretchy="false" form="prefix">(</mo><mi>D</mi><mo>,</mo><mi>&mu;</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">L^{2}(D, \mu)</annotation></semantics></math>
 functions on the unit
-<math role="math" aria-label="\operatorname{disc} D=\{z \in \mathcal{C}:|z|&lt;1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">disc</mi><mo>&#8289;</mo></mrow><mi>D</mi><mo>=</mo><mo stretchy="false" form="prefix">{</mo><mi>z</mi><mo>&isin;</mo><mi>𝒞</mi><mo>:</mo><mo stretchy="false" form="prefix">|</mo><mi>z</mi><mo stretchy="false" form="prefix">|</mo><mo>&lt;</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\operatorname{disc} D=\{z \in \mathcal{C}:|z|&lt;1\}</annotation></semantics></math>,
+<math role="math" aria-label="\operatorname{disc} D=\{z \in \mathcal{C}:|z|&lt;1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">disc</mi><mo>&#8289;</mo></mrow><mi>D</mi><mo>=</mo><mo stretchy="false" form="prefix">{</mo><mi>z</mi><mo>&isin;</mo><mi>&Cscr;</mi><mo>:</mo><mo stretchy="false" form="prefix">|</mo><mi>z</mi><mo stretchy="false" form="prefix">|</mo><mo>&lt;</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\operatorname{disc} D=\{z \in \mathcal{C}:|z|&lt;1\}</annotation></semantics></math>,
 with
 <math role="math" aria-label="\mu" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>&mu;</mi><annotation encoding="application/x-tex">\mu</annotation></semantics></math>
 two-dimensional measure,