Gordon Morrison
02/24/2024 · Elementary School
4.a. Let \( \left\{\mathrm{g}_{\mathrm{n}}\right\} \) be a sequence of measurable functions on \( [0,1][0,1] \) such that \( \lg _{\mathrm{n}}(\mathrm{x}) \mid \leq \mathrm{n} 1 \) for all x and n . Show that \( \lim _{n \rightarrow \infty} \mathrm{~g}_{\mathrm{n}} \mathrm{d}=0 \) using the Dominated Convergence Theorem. ( 8 Marks) b. i. Determine the Lebesgue measure of set \( \mathrm{E}=(1,2) \mathrm{U}(2,3) \quad \) (2 Marks)
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a. To prove \( \lim _{n \rightarrow \infty} \mathrm{~g}_{\mathrm{n}} \mathrm{d}=0 \) using the Dominated Convergence Theorem, show that \( \{g_n\} \) is measurable, dominated by an integrable function \( M \), converges pointwise to 0, and then apply the theorem.
b. i. The Lebesgue measure of set \( E = (1,2) \cup (2,3) \) is 2.
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