Probability with Measure
2.4 Exercises on Chapter 2
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2.1 Let \((a_n)\) be a sequence of extended reals. Show that \((a_n)\) is bounded if and only if we have \(-\infty <\liminf _n a_n\leq \limsup _n a_n<\infty \).
Hint: Recall from real analysis that sequences which converge in \(\R \) are necessarily bounded.
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2.2 Let \(f_n:\R \to \R \) by \(f_n(x)=n\1_{[0,\frac 1n]}(x)\). Show that \(f_n\to 0\) almost everywhere.
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2.3 Let \((S, \Sigma , m)\) be a measure space. Let \(f,g\) and, for each \(n\in \N \), \(f_n\) and \(g_n\) be functions from \(S\) to \(\R \). Suppose that \(f_n\to f\) almost everywhere, and \(g_n\to g\) almost everywhere. Show that \(f_{n} + g_{n} \rightarrow f + g\) almost everywhere, and \(f_{n}g_{n} \rightarrow fg\) almost everywhere.
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(a) For each \(n\in \N \) let \(a_n,b_n\in [0,\infty ]\). Show that
\[\sum _{n=1}^\infty a_n + \sum _{n=1}^\infty b_n=\sum _{n=1}^\infty (a_n+b_n).\]
If \(a_n,b_n\in \R \) and the summations converge in \(\R \), implying absolute convergence because all terms are non-negative, then you already know this. The point of this question is to work in \(\ov {\R }\).
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(b) Let \(m\) and \(n\) be measures on \((S, \Sigma )\). Deduce that \(m + n\) is a measure on \((S, \Sigma )\), where \((m + n)(A) = m(A) + n(A)\) for all \(A \in \Sigma \).
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(c) Let \(S = \{x_{1}, x_{2}, \ldots , x_{n}\}\) be a finite set and \(c_{1}, c_{2}, \ldots , c_{n}\) be non-negative numbers. Let
\[m = \sum _{i=1}^{n}c_{i}\de _{x_{i}}.\]
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(i) Show that \(m\) is a measure on \((S, \mc {P}(S))\).
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(ii) What condition should be imposed on \(\{c_{1}, c_{2}, \ldots , c_{n}\}\) for \(m\) to be a probability measure?
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(a) Let \(A, B \sw S\). Show that:
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(i) \({\1}_{A \cup B} = {\1}_{A} + {\1}_{B} - {\1}_{A \cap B}\).
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(ii) \({\1}_{A \cap B} = {\1}_{A}{\1}_{B}\).
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(iii) If \(B\sw A\) then \({\1}_{A \sc B} = {\1}_{A} - {\1}_{B}\).
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(b) Let \((A_{n})\) be a sequence of disjoint subsets of \(S\) and let \(A = \bigcup _{n=1}^{\infty }A_{n}\). Explain how the function \(\sum _{n=1}^{\infty } {\1}_{A_{n}}\) is defined and show that \({\1}_{A} = \sum _{n=1}^{\infty } {\1}_{A_{n}}\).
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2.6 Let \((a_n)\) and \((b_n)\) be a sequences of extended reals.
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(a) Show that:
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(i) \(\limsup _{n} a_n + \limsup _{n} b_n \leq \limsup _{n}\,(a_n+b_n)\)
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(ii) \(\l (\limsup _{n} a_n\r )\l (\limsup _{n} b_n\r ) \leq \limsup _{n}\,a_nb_n\)
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(iii) \(c\limsup _{n} a_n = \limsup _{n}\,(ca_n)\) for \(c\in [0,\infty )\)
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(iv) if \(a_n\leq b_n\) for all \(n\), then \(\limsup _n a_n \leq \limsup _n b_n\).
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(b) Derive similar relationships for \(\liminf \).
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2.7 Let \(f_n:\R \to \R \) be given by \(f_n(x)=e^{-nx^2}\) and let \(f:\R \to \R \) be given by \(f(x)=\1_{\{0\}}(x)\). Show that \(f_n \to f\) pointwise but not uniformly.
Challenge questions
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2.8 This question involves metric spaces and is off-syllabus for that reason. \((\star )\)
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(a) Show that \((\ov {\R },d)\) is a metric space, where \(d(x,y)=|\arctan (x)-\arctan (y)|\). Here we set \(\arctan (-\infty )=-1\) and \(\arctan (\infty )=1\).
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(b) Show that \((\ov {\R },d)\) is compact.
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(c) Let \((a_n)\) be a sequence of extended real numbers. Show that the following are equivalent:
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(i) \(a_n\to a\)
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(ii) If \((a_{r_n})\) is a convergent subsequence of \((a_n)\) then \(a_{r_n}\to a\).
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2.9 Let \((a_n)\) be any sequence within \(\overline {\R }\) and let
\[\mathscr {L}=\{a\in \ov {\R }\-\text { there exists a subsequence }(a_{r_n})\text { of }(a_n)\text { with }a_{r_n}\to a\}.\]
Show that \(\liminf _n a_n = \inf \mathscr {L}\) and \(\limsup _n a_n=\sup \mathscr {L}.\)
Part (c) of Exercise 2.8 will help. If you prefer to avoid using that, you should impose the extra condition that \(a_n\in \R \) is a bounded sequence and use 2.1.