Probability with Measure

4.11 Exercises on Chapter 4

On integrals of simple functions and non-negative functions

• 4.1 Let \(f: \R \rightarrow \R \) be defined as follows

  \[ f(x) = \begin {cases} 2 & \text { if }x\in [-2,-1]\\ -1 & \text { if }x\in (-1,1) \\ 3 & \text { if }x\in [1,2) \\ -5 & \text { if }x\in [2,3). \end {cases} \]

  • (a) Write \(f\) explicitly as a simple function and calculate \(\int _{\R }f(x)\,dx\).

  • (b) Write down \(f_{+}\) and \(f_{-}\) and confirm that they are non-negative simple functions. Calculate \(\int _{\R }f_+(x)\,dx\) and \(\int _{\R }f_-(x)\,dx\) and check that \(\int _{\R }f_+(x)\,dx-\int _{\R }f_-(x)\,dx=\int _{\R }f(x)\,dx\).

• 4.2 Let \((S, \Sigma , m)\) be a measure space, \(A \in \Sigma \) and \(f:S\to \R \) be a simple function. Show that \({\1}_{A}f\) is also a simple function.

• 4.3 Use Lemma 4.2.3 (Markov’s inequality) to prove the following version of Chebychev’s inequality. If \(f:S \rightarrow \R \) is a measurable function and \(c > 0\) then

  \[ m\big (\{x \in S; |f(x)| \geq c\}\big ) \leq \frac {1}{c^{2}}\int _{S}f^{2}\, dm. \]

  Formulate and prove a similar inequality where \(c^{2}\) is replaced by \(c^{p}\) for \(p \geq 1\).

• 4.4 This question contributes to the proof of Theorem 4.5.3. Let \(f,g\in L^1(S,\Sigma ,m)\) and let \(\alpha \in \R \). Note that the integrals of \(f\) and \(g\) are defined by (4.15). Using the properties of integrals of non-negative functions that were proved in Sections 4.2 and 4.3, show that:

  • (a) \(\left |\int _{S}f \,dm\right | \leq \int _{S}|f|\,dm\)

  • (b) \(\int _{S}|f + g|\,dm \leq \int _{S}|f|\,dm + \int _{S}|g|\,dm\)

  • (c) \(\alpha \int _S f\,dm = \int _S \alpha f\,dm\)

    Hint: First consider \(\alpha \geq 0\), then \(\alpha =-1\) and then combine to handle \(\alpha <0\).

  • (d) If \(f\leq g \) then \(\int _S f\,dm \leq \int _S g\,dm\).

    In part (d) you may use linearity (of the full Lebesgue integral) because parts (b) and (c) provide the missing pieces that complete the proof of linearity.

On \(\Lone \) and the convergence theorems

• 4.5 Let \((S,\Sigma ,m)\) be a measure space and suppose that \(m\) is a finite measure. Suppose that \(f:S\to \R \) is bounded. Show that \(f\in \Lone \).

• 4.6 Determine if the function \(g:(0,1)\to \R \) by \(g(x)=\log x\) is in \(\mc {L}^1\).

• 4.7 Consider the sequence \((f_{n})\) on the measure space \((\R , {\cal B}(\R ), \lambda )\) where \(f_{n} = n{\1}_{(0, 1/n)}\). Show that \((f_{n})\) converges pointwise to zero, but that \(\int _\R f_n\,d\lambda =1\) for all \(n\in \N \).

  Does either of the monotone or dominated convergence theorems apply to this situation?

• 4.8 Show that if \(f\in \Lone _\R \) then so is the mapping \(x \rightarrow \cos (\alpha x)f(x)\), where \(\alpha \in \R \). Prove that

  \[ \lim _{n \rightarrow \infty }\int _{\R }\cos (x/n)f(x)dx = \int _{\R }f(x)dx.\]

• 4.9 Let \((S, \Sigma , m)\) be a measure space and \((A_{n})\) be a sequence of disjoint sets with \(A_{n} \in \Sigma \) for each \(\nN \). Set \(A = \bigcup _{n=1}^{\infty }A_{n}\) and let \(f: S \rightarrow \R \) be measurable. Show that

  \[\int _A |f|\,dm=\sum _{n=1}^\infty \int _{A_n}|f|\,dm.\]

  Hint: Use the monotone convergence theorem.

• 4.10 Let \((a_n)_{n\in \N }\) be a real valued sequence, viewed as a function \(a:\N \to \R \) with \(a_n=a(n)\). We work over the measure space \((\N ,\mc {P}(\N ),\#)\), where \(\#\) denotes counting measure.

  • (a) Suppose that \(a_n\geq 0\) and fix \(N\in \N \). Let \(a^{(N)}_n=\1_{\{n\leq N\}} a_n\). Show that \(a^{(N)}\) is a simple function, write down its integral, and use the monotone convergence theorem to deduce that

    \(\seteqnumber{0}{4.}{27}\)

    \begin{equation} \label {eq:ps_series_int} \int _\N a\,d\#=\sum \limits _{n=1}^\infty a_n. \end{equation}

  • (b) Now consider a general \(a=(a_n)_{n\in \N }\sw \R \). Explain briefly why \(a\in \Lone (\N )\) if and only if \(\sum _n|a_n|<\infty \) and deduce that (4.28) holds in this case too.

• 4.11 Show that \(f\eqae g\) defines an equivalence relation on the set of all real-valued measurable functions defined on \((S, \Sigma , m)\).

• 4.12 \((\star )\) Prove the reverse Fatou lemma: if \((f_{n})\) is a sequence of non-negative measurable functions for which \(f_{n} \leq f\) for all \(\nN \), where \(f\in \Lone \), then

  \[ \ls \int _{S}f_{n}\,dm \leq \int _{S}\ls f_{n}\,dm.\]

  Hint: Apply Lemma 4.6.3 to \(f - f_{n}\).

On integration of complex valued functions

• 4.13 \((\Delta )\) Write down a version of the dominated convergence theorem applicable to functions \(f:S\to \C \). Prove it using the real case.

• 4.14 \((\Delta )\) Let \(a\in \R \). Calculate the value of \(\int _0^x e^{i a y}\,dy\).

• 4.15 \((\Delta )\) Of Exercises 4.5, 4.9, 4.10 and 4.11, which of these results have natural extensions to complex valued functions? Justify your answers briefly.

  You will need to solve those exercises first!

Challenge questions

• 4.16 Let \((S, \Sigma , m)\) be a measure space and \(f: [a, b] \times S \rightarrow \R \) be a measurable function for which

  • (i) The mapping \(x \rightarrow f(t,x)\) is in \(\mc {L}^1\) for all \(t \in [a,b]\),

  • (ii) The mapping \(t \rightarrow f(t,x)\) is continuous for all \(x \in S\),

  • (iii) There exists \(g\in \Lone \) such that \(|f(t,x)| \leq g(x)\) for all \(t \in [a,b], x \in S.\)

  Use the dominated convergence theorem to show that the mapping \(t \rightarrow \int _{S}f(t,x)\,dm(x)\) is continuous at all \(t\in [a, b]\).

  Hint: Use continuity in terms of sequences, that is show that \(\lim _{n \rightarrow \infty }\int _{S}f(t_{n},x)\,dm(x) = \int _{S}f(t,x)\,dm(x)\) for any sequence \((t_{n})\) satisfying \(\lim _{n \rightarrow \infty }t_{n} = t\).

• 4.17 Let \((S, \Sigma , m)\) be a measure space and \(f: [a, b] \times S \rightarrow \R \) be a measurable function for which

  • (i) The mapping \(x \rightarrow f(t,x)\) is in \(\mc {L}^1\) for all \(t \in [a,b]\),

  • (ii) The mapping \(t \rightarrow f(t,x)\) is differentiable for all \(x \in S\),

  • (iii) There exists \(h\in \Lone \) such that \(\ds \left |\frac {\partial f(t,x)}{\partial t}\right | \leq h(x)\) for all \(t \in [a,b], x \in S.\)

  Show that the mapping \(t \rightarrow \int _{S}f(t,x)\,dm(x)\) is differentiable on \((a, b)\) and that

  \[ \frac {\p }{\p t}\int _{S}f(t,x)\,dm(x) = \int _{S}\frac {\partial f(t,x)}{\partial t}\, dm(x).\]

  Hint: Use the mean value theorem.

• 4.18 Let

  \(\seteqnumber{0}{4.}{28}\)

  \begin{align*} f(x) &= -2xe^{-x^{2}}\\ f_{n}(x) &= \sum _{r=1}^{n}\l (-2r^{2}xe^{-r^{2}x^{2}} + 2(r+1)^{2}xe^{-(r+1)^{2}x^{2}}\r ) \end{align*} for all \(x\in \R \).

  • (a) Show that \(f(x) = \lim _{n \rightarrow \infty }f_{n}(x)\) for all \(x\in \R \).

  • (b) Let \(a>0\). Show that \(f\) and \(f_{n}\) are Riemann integrable over \([0, a]\) for all \(\nN \) but that

    \[\int _{0}^{a}f(x)\,dx \neq \lim _{n \rightarrow \infty }\int _{0}^{a}f_{n}(x)\,dx.\]

  Neither the monotone or dominated convergence theorems can be used here (follow up exercise: explain why not). This example illustrates that things can go badly wrong without them, even when \(f_n(x)\to f(x)\) for all \(x\).

Additional questions \((\star )\)

These questions explore the definition and properties of the Fourier transform. They are off syllabus but you may find them interesting. They involve integration in \(\C \), as described in Section 4.8, and you will need extensions of several key results (e.g. linearity, dominated convergence) to that setting.

• 4.19 Let \(f:\R \to \R \). If \(f\in \Lone (\R , {\mathcal B}(\R ), \lambda )\), where \(\lambda \) is Lebesgue measure, define its Fourier transform \(\widehat {f}(y)\) for each \(y \in \R \), by

  \[\begin {aligned} \widehat {f}(y) & = \int _{\R }e^{-ixy}f(x) dx \\ & = \int _{\R }\cos (xy)f(x)dx - i \int _{\R }\sin (xy)f(x)dx. \end {aligned}\]

  Prove that \(|\widehat {f}(y)| < \infty \) and so \(\widehat {f}\) is a well-defined function from \(\R \) to \(\C \). Show also that the Fourier transformation \({\mathcal F}f =\widehat {f}\) is linear, i.e. for all \(f, g\in \Lone \) and \(a, b \in \R \) we have

  \[ \widehat {af + bg} = a \widehat {f} = b\widehat {g}. \]

• 4.20 Recall Dirichlet’s jump function \({\1}_{\mathbb {Q}}\). Does it make sense to write down the Fourier coefficients \(a_{n} = \frac {1}{\pi }\int _{-\pi }^{\pi }{\1}_{\mathbb {Q}}(x)\cos (nx)dx\) and \(b_{n} = \frac {1}{\pi }\int _{-\pi }^{\pi }{\1}_{\mathbb {Q}}(x)\sin (nx)dx\) as Lebesgue integrals? If so, what values do they have? Can you associate a Fourier series to \({\1}_{\mathbb {Q}}\)? If so, (and if it is convergent) what does it converge to?

• 4.21 Fix \(a \in \R \) and define the shifted function \(f_{a}(x) = f(x - a)\). If \(f\in \Lone \), show that \(f_{a}\in \Lone \), and deduce that \(\widehat {f_{a}}(y) = e^{-iay}\widehat {f}(y)\) for all \(y \in \R \).

• 4.22 Show that the mapping \(y \rightarrow \widehat {f}(y)\) is continuous from \(\R \) to \(\C \).

• 4.23 Suppose that the mappings \(x \rightarrow f(x)\) and \(x \rightarrow xf(x)\) are both in \(\Lone \). Show that \(y \rightarrow \widehat {f}(y)\) is differentiable and that for all \(y \in \R \),

  \[ (\widehat {f})^{\prime }(y) = -i\widehat {g}(y),\]

  where \(g(x) = xf(x)\) for all \(x \in \R \).

  Hint: Use the inequality \(|e^{ib} - 1| \leq |b|\) for \(b \in \R \).

• 4.24 Assume that \(f,g\in \Lone (\R ,\mc {B}(\R ),\lambda )\) and that \(g\) is bounded. Define the convolution \(f * g\) of \(f\) with \(g\) by

  \[ (f * g)(x) = \int _{\R }f(x - y)g(y)dy,\]

  for all \(x \in \R \). Show that \(|(f*g)(x)| < \infty \), and so \(f *g\) is a well–defined function from \(\R \) to \(\R \). Show further that \(f * g\in \Lone \), and that the Fourier transform of the convolution is the product of the Fourier transforms, i.e. that for all \(y \in \R \),

  \[ \widehat {f * g}(y) = \widehat {f}(y)\widehat {g}(y).\]