Chapter 7: The Fatou Lemmas
- Page ID
- 208227
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Since integrals can be thought of as limits of sums, the finite subadditivity of limit superior and finite superadditivity of limit inferior carry forward to Lebesgue delta integrals. These generalisations are called the Fatou lemmas.
Suppose that \(\{f_n\}_{n\in \mathbb{N}}\) is a sequence of Lebesgue delta measurable functions \(f_n: \mathbb{T}\to [0, \infty]\). Then
\[ \int\limits_{\mathbb{T}}\liminf\limits_{n\to\infty}f_n\Delta \mu_\Delta\leq \liminf\limits_{n\to\infty} \int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta. \notag\]
In particular, if \(\{f_n\}_{n\in \mathbb{N}}\) is a sequence of nonnegative Lebesgue delta measurable functions and \(f_n\to f\) as \(n\to \infty\) \(\mu_\Delta\)-a.e., then
\[ \int\limits_{\mathbb{T}}f\Delta\mu_\Delta\leq \liminf\limits_{n\to\infty}\int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta. \notag\]
Proof
Define
\[ g_n(t)=\inf\limits_{m\geq n}f_m(t),\quad t\in \mathbb{T},\quad n\in \mathbb{N}. \notag\]
Then \(\{g_n\}_{n\in \mathbb{N}}\) is a pointwise increasing sequence and
\[ \lim\limits_{n\to\infty} g_n= \liminf\limits_{n\to\infty} f_n. \notag\]
We apply the Monotone Convergence Theorem to the sequence \(\{g_n\}_{n\in \mathbb{N}}\) to get
\[ \begin{aligned} \lim\limits_{n\to\infty}\int\limits_{\mathbb{T}}g_n\Delta\mu_\Delta=& \int\limits_{\mathbb{T}}\lim\limits_{n\to\infty} g_n\Delta \mu_\Delta\\ =& \int\limits_{\mathbb{T}}\liminf\limits_{n\to\infty}f_n \Delta\mu_\Delta. \end{aligned} \notag\]
On the other hand, we have
\[ 0\leq g_n\leq f_m,\quad m\geq n. \notag\]
Then
\[ \int\limits_{\mathbb{T}}g_n\Delta\mu_\Delta\leq \int\limits_{\mathbb{T}}f_m\Delta\mu_\Delta,\quad m\geq n. \notag\]
Thus, we have
\[ \int\limits_{\mathbb{T}}g_n\Delta\mu_\Delta\leq \inf\limits_{m\geq n}\int\limits_{\mathbb{T}}f_m\Delta\mu_\Delta, \notag\]
whereupon
\[ \lim\limits_{n\to\infty}\int\limits_{\mathbb{T}}g_n\Delta\mu_\Delta\leq \liminf\limits_{n\to\infty}\int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta \notag\]
and
\[ \int\limits_{\mathbb{T}}\liminf\limits_{n\to\infty} f_n\Delta\mu_\Delta\leq \liminf\limits_{n\to\infty}\int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta, \notag\]
which completes the proof.
Suppose that \(\{f_n\}_{n\in \mathbb{N}}\) is a sequence of Lebesgue delta measurable functions \(f_n: \mathbb{T}\to [0, \infty]\) such that there exists a nonnegative Lebesgue delta measurable function \(g\) for which
\[ f_n\leq g,\quad n\in \mathbb{N}. \notag\]
Then
\[ \limsup\limits_{n\to\infty} \int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta\leq \int\limits_{\mathbb{T}}\limsup\limits_{n\to\infty}f_n\Delta \mu_\Delta. \notag\]
Proof
Define
\[ h_n=g-f_n,\quad n\in \mathbb{N}. \notag\]
Applying the Fatou Lemma to the sequence \(\{h_n\}_{n\in \mathbb{N}}\), we find
\[ \int\limits_{\mathbb{T}}\liminf\limits_{n\to\infty}(g-f_n)\Delta\mu_\Delta\leq \limsup\limits_{n\to\infty}\int\limits_{\mathbb{T}}(g-f_n)\Delta\mu_\Delta, \notag\]
or
\[ \int\limits_{\mathbb{T}}g\Delta\mu_\Delta +\int\limits_{\mathbb{T}}\liminf\limits_{n\to\infty}(-f_n)\Delta\mu_\Delta \leq \int\limits_{\mathbb{T}}g\Delta\mu_\Delta +\liminf\limits_{n\to\infty}\left(-\int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta\right), \notag\]
or
\[ -\int\limits_{\mathbb{T}}\limsup\limits_{n\to\infty}f_n\Delta\mu_\Delta\leq -\limsup\limits_{n\to\infty}\int\limits_{\mathbb{T}}f_n\Delta\mu_\Delta, \notag\]
whereupon we get the desired result.
Let \(f, g: \mathbb{T}\to [0, \infty]\) be nonnegative Lebesgue delta measurable functions and \(\alpha, \beta\geq 0\). Then
\[ \int\limits_{\mathbb{T}}(\alpha f+\beta g)\Delta\mu_\Delta=\alpha\int\limits_{\mathbb{T}}f\Delta\mu_\Delta+\beta \int\limits_{\mathbb{T}}g\Delta \mu_\Delta. \notag\]
Proof
Firstly, we will prove that \begin{equation} \label{51}\int\limits_{\mathbb{T}}(f+g)\Delta\mu_\Delta=\int\limits_{\mathbb{T}}f\Delta\mu_\Delta+\int\limits_{\mathbb{T}}g\Delta\mu_\Delta. \end{equation} Consider the pointwise increasing sequences of simple functions \(\{\phi_n\}_{n\in \mathbb{N}}\) and \(\{\psi_n\}_{n\in \mathbb{N}}\) such that \(\phi_n\to f\) and \(\psi_n\to g\) as \(n\to\infty\). Then
\[ \phi_n+\psi_n\to f+g\quad \mbox{as}\quad n\to\infty. \notag\]
By the Monotone Convergence Theorem, we get
\[ \begin{aligned} \int\limits_{\mathbb{T}}(f+g)\Delta\mu_\Delta=& \lim\limits_{n\to\infty}I(\phi_n+\psi_n)\\ =& \lim\limits_{n\to\infty}(I(\phi_n)+I(\psi_n))\\ =& \lim\limits_{n\to\infty}I(\phi_n)+\lim\limits_{n\to\infty}I(\psi_n)\\ =& \int\limits_{\mathbb{T}} f\Delta\mu_\Delta+\int\limits_{\mathbb{T}}g\Delta\mu_\Delta. \end{aligned} \notag\]
Now, we apply \eqref{51} for the functions \(\alpha f\) and \(\beta g\) and using the homogeneity of the Lebesgue delta integral, we find
\[ \begin{aligned} \int\limits_{\mathbb{T}}((\alpha f)+(\beta g))\Delta\mu_\Delta=& \int\limits_{\mathbb{T}}\alpha f\Delta\mu_\Delta+\int\limits_{\mathbb{T}}\beta g\Delta\mu_\Delta\\ =& \alpha \int\limits_{\mathbb{T}}f \Delta\mu_\Delta+\beta \int\limits_{\mathbb{T}}g\Delta \mu_\Delta. \end{aligned} \notag\]
This completes the proof.