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Author: lenis2000

graduate/exams/analysis/2023Jan_real.html

@@ -246,7 +246,7 @@ <h2 class="unnumbered" id="problem-3.">Problem 3.</h2>
 <p>for all
 <math role="math" aria-label="E \in \Sigma" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>&isin;</mo><mi mathvariant="normal">&Sigma;</mi></mrow><annotation encoding="application/x-tex">E \in \Sigma</annotation></semantics></math>.<br />
 (i) Show that
-<math role="math" aria-label="0 \leq f \leq 1 \lambda" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&le;</mo><mi>f</mi><mo>&le;</mo><mn>1</mn><mi>&lambda;</mi></mrow><annotation encoding="application/x-tex">0 \leq f \leq 1 \lambda</annotation></semantics></math>-a.e.<br />
+<math role="math" aria-label="0 \leq f \leq 1 \,\lambda" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&le;</mo><mi>f</mi><mo>&le;</mo><mn>1</mn><mspace width="0.25em"/><mi>&lambda;</mi></mrow><annotation encoding="application/x-tex">0 \leq f \leq 1 \,\lambda</annotation></semantics></math>-a.e.<br />
 (ii) If
 <math role="math" aria-label="F=\{x: f(x)=1\}" display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo>=</mo><mo stretchy="false" form="prefix">{</mo><mi>x</mi><mo>:</mo><mi>f</mi><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo stretchy="false" form="postfix">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">F=\{x: f(x)=1\}</annotation></semantics></math>,
 show that