Chapter 2 Integration Theory
2.1 The lebesgue integral: basic properties and convergence theorems
2.1.1 Stage one: simple functions
Proposition 2.1 The integral of simpe functions defined bve satifies the following properties:
- Independence of the representation. If \(\varphi=\sum_{k=1}^Na_k\chi_{E_k}\) is any representation of \(\varphi\), then \[ \int \varphi=\sum_{k=1}^Na_km(E_k). \]
- Linearity.
- Additivity.
- Monotonicity.
- Triangle inequality. If \(\varphi\) is a simple function, then so is \(|\varphi|\), and \[ \left|\int\varphi\right|\le \int|\varphi|. \]
2.1.2 Stage two: bounded functions supported on a set of finite measure
Lemma 2.1 Let \(f\) be a bounded function supported on a set \(E\) of finite measure. If \(\{\varphi_n\}_{n=1}^\infty\) is any sequence of simple functions bounded by \(M\), supported on \(E\), and with \(\varphi_n(x)\to f(x)\) for a.e. \(x\), then:
- The limit \(\lim_{n\to\infty}\int\varphi_n\) exists.
- If \(f=0\) a.e., then the limit \(\lim_{n\to\infty}\int\varphi_n\) equals 0.
Proof.
Setting \(I_n=\int\varphi_n\) and applying Egorov’s theorem which is proven in Chapter 1 we have that for and large \(n\) and \(m\) \[ \begin{aligned} |I_n-I_m| &\le \int_E|\varphi_n-\varphi_m|\\ &=\int_{A_\epsilon}|\varphi_n-\varphi_m|+\int_{E-A_\epsilon}|\varphi_n-\varphi_m|\\ &\le \int_{A_\epsilon}\epsilon\ dx+\int_{E-A_\epsilon}2M\ dx\\ &\le m(E)\epsilon+2M\epsilon. \end{aligned} \] given any \(\epsilon>0\). This proves that \(\{I_n\}\) os a Cauchy sequence nd hence converges. If \(f=0\), letting \(m\) tend to infinity we have \(|I_n-f|=|I_n|\le m(E)\epsilon+2M\epsilon\), which yields \(\lim_{n\to\infty}I_n=0\).For a bounded function \(f\) that is supported on sets of finite measure, we define its Lebesgue integral by \[ \int f=\lim_{n\to\infty}\int\varphi_n. \] where \(\{\varphi_n\}\) is any sequence of simple functions satisfying: \(|\varphi_n|\le M\), each \(\varphi_n\) is supported on the support of \(f\), and \(\varphi_n(x)\to f(x)\) for a.e. \(x\) as \(n\) tends to infinity.
Next, we must show that \(\int f\) is independent of the limiting sequence \(\{\varphi_n\}\) used, in order for the integral to be well-defined. Suppose that \(\{\psi_n\}\) is another sequence of simple functions that satisfies the properties above. Then, if \(\eta_n = \varphi_n- \psi_n\), the sequence \(\{\eta_n\}\) consists of simple functions bounded by \(2M\), supported on a set of finite measure, and such that \(\eta_n\to 0\) a.e. as \(n\) tends to infinity. Applying the lemma we find \[ \lim_{n\to\infty}\int \varphi_n =\lim_{n\to\infty}\int \psi_n+\lim_{n\to\infty}\int \eta_n=\lim_{n\to\infty}\int \psi_n \] as desired.
Proposition 2.2 Suppose \(f\) and \(g\) are bounded functions supported on sets of finite measure. Then the following properties hold.
- Linearity.
- Additivity.
- Monotonicity.
- Triangle inequality.
Theorem 2.1 (Bounded convergence theorem)
Suppose that \(\{f_n\}\) is a sequence of measurable functions that are all bounded by \(M\), are supported on a set \(E\) of finite measure, and \(f_n(x)\to f(x)\) a.e. \(x\) as \(n\to\infty\). Then \(f\) is measurable, bounded ,supported on \(E\) for a.e. \(x\), and \[ \int|f_n-f|\to 0 \text{ as } n\to\infty. \] Consequently, \[ \int f_n\to \int f \text{ as } n\to\infty. \]Proof.
The proof is a reprise of the argument in Lemma 2.1. Given \(\epsilon>0\), we may find, by Egorov’ theorem, \[ \begin{aligned} \int|f_n-f| &\le \int_{A_\epsilon}|f_n-f|+\int_{E-A_\epsilon}|f_n-f|\\ &\le m(E)\epsilon+2M\epsilon. \end{aligned} \] for all large \(n\).2.1.3 Return to Riemann integrable functions
Theorem 2.2
Suppose \(f\) os Riemann integrable on the closed interval \([a,b]\). Then \(f\) is measurable, and \[ \cal \int_{[a,b]}^Rf(x)\ dx=\int_{[a,b]}^Lf(x)\ dx. \]Proof.
By definition of Riemann integrability, \(f\) is bounded, say \(|f(x)|\le M\), and we may construct two consequences of step functions \(\{\varphi_k\}\) and \(\{\psi_k\}\) that satisfy the following properties: \(|\varphi_k(x)\le M|\) and \(|\psi_k(x)\le M|\) for all \(x\in[a,b]\) and \(k\ge 1\), \[ \varphi_1(x)\le \varphi_2(x) \le \cdots\le f\le\cdots\le \psi_2(x)\le \psi_1, \] and \[ \lim_{k\to\infty}\cal \int_{[a,b]}^R\varphi_k(x)\ dx=\lim_{k\to\infty}\cal \int_{[a,b]}^R\psi_k(x)\ dx=\cal \int_{[a,b]}^Rf(x)\ dx. \] Notice that \[ \cal \int_{[a,b]}^R\varphi_k(x)\ dx=\int_{[a,b]}^L\varphi_k(x)\ dx, \] and \[ \cal \int_{[a,b]}^R\psi_k(x)\ dx=\cal \int_{[a,b]}^L\psi_k(x)\ dx. \] Consider Theorem 2.1 (Bounded convergence theorem) and you will complete the proof.2.1.4 Stage three: non-negative functions
In the case of such a function \(f\) we define its Lebesgue integral by \[ \int f =\sup_g\int g. \]
Proposition 2.3 The integral of non-negative measurable functions enjoys the following properties:
- Linearity.
- Additivity.
- Monotonicity.
- If \(g\) is integrable and \(0\le f\le g\), then \(f\) is integrable.
- If \(f\) is integrable, then \(f(x)<\infty\) for a.e. \(x\).
- If \(\int f=0\), then \(f(x)=0\) for a.e. \(x\).
Proof.
We just prove the first assertion. Let \(\varphi\), \(\psi\) and \(\eta\) be non-negative functions bounded and supported on sets of finite measure, where \(\varphi\le f\), \(\psi\le g\) and \(\eta\le f+g\), then \(\varphi+\psi\le f+g\). Consequently, \[ \int f+ \int g\le \int (f+g). \] On the other hand, if we define \(\eta_1=\min(f(x),\eta(x))\) and \(\eta_2=\eta-\eta_1\), then \[ \int \eta=\int(\eta_1+\eta_2)=\int \eta_1+\int \eta_2\le\int f+ \int g, \] which means that \[ \int (f+g)\le \int f+ \int g. \]
Lemma 2.2 (Fatou)
Suppose \(\{f_n\}\) is a sequence of measurable functions with \(f_n\ge 0\). If \(\lim_{n\to\infty}f_n(x)=f(x)\) for a.e. \(x\), then \[ \int f\leq \liminf_{n\to\infty}\int f_n. \]Proof.
Suppose \(0\le g\le f\), where \(g\) is bounded and supported on a set \(E\) of finite measure. If we set \(g_n(x)=\min(g(x),f_n(x))\), then \(g_n\to g\) a.e. as \(n\to \infty\). By Theorem 2.1 (Bounded convergence theorem) we have \[ \int g_n\to \int g. \] Since \(g_n\le f_n\), we have \(\int f_n\le \int g_n\), so that \[ \int g\le \liminf_{n\to\infty}\int f_n. \] Taking the supremum over all \(g\) yields the desired inequality.Corollary 2.1 Suppose \(f\) is a non-negative measurable function, and \(\{f_n\}\) a sequence of non-negative measurable functions with \(f_n(x) \le f(x)\) and \(f_n(x)\to f(x)\) for a.e. \(x\). Then \[ \lim_{n\to\infty}\int f_n=\int f. \]
Corollary 2.2 (Monotone convergence theorem) Suppose \(\{f_n\}\) a sequence of non-negative measurable functions with \(f_n(x) \nearrow f(x)\). Then \[ \lim_{n\to\infty}\int f_n=\int f. \]
Corollary 2.3 Consider a series \(\sum_{k=1}^{\infty}a_k(x)\), where \(a_k(x)\ge 0\) is measurable for every \(k\ge 1\). Then \[ \int \sum_{k=1}^{\infty}a_k(x)\ dx =\sum_{k=1}^{\infty}\int a_k(x)\ dx. \]
2.1.5 General form
In this case, we define the Lebesgue integral of \(f\) by \[ \int f =\int f^+-\int f^-. \]
Proposition 2.4 The integral of Lebesgue integrable functions is linear, additive, monotonic, and satisfies the triangle inequality.
Proposition 2.5 Suppose \(f\) is integral on \(\mathbb{R}^d\). Then for every \(\epsilon>0\):
- There exists a set of finite measure \(B\) (a ball, for example) such that \[ \int_{B^c}|f|<\epsilon. \]
- There is a \(\delta>0\) such that
\[
\int_E|f|<\epsilon\quad \text{ whenever }m(E)<\delta.
\]
Proof.
Assume that \(f\ge 0\): 1. \(B_N=(-N,N)\); 2. \(E_N=\{x:f(x)\le N\}\).
Theorem 2.3 (Dominated convergence theorem)
Suppose \(\{f_n\}\) is a sequence of measurable functions such that \(f_n\to f\) a.e. \(x\) as \(n\) tends to infinity. If \(f_n(x)\le g(x)\), where \(g\) is integrable, then \[ \int |f_n-f|\to 0 \quad \text{ as }n\to \infty, \] and consequently \[ \int f_n \to \int f. \]Proof.
For each \(N\ge 0\) let \(E_N=\{x:|x|\le N, g(x)\le N\}\). \[ \begin{aligned} \int|f_n-f|&=\int_{E_N}|f_n-f|+\int_{E_N^c}|f_n-f|\\ &\le \int_{E_N}|f_n-f|+2\int_{E_N^c}g\\ &\le \epsilon +2\epsilon \end{aligned} \] for all large \(n\). This prove the theorem.2.2 The space \(L^1\) of integrable functions
For any integrable function \(f\) on \(\mathbb{R}^d\) we define the \(L^1\)-norm of \(f\), \[ \|f\|=\|f\|_{L^1}=\|f\|_{L^1(\mathbb{R}^d)}=\int_{\mathbb{R}^d}|f|. \]
Proposition 2.6 Suppose \(f\) and \(g\) are two functions in \(L^1(\mathbb{R}^d)\).
- \(\|af\|=|a|\|f\|\) for all \(a\in\mathbb{C}\).
- \(\|f+g\|\le \|f\|+\|g\|\).
- \(\|f\|=0\) iff \(f=0\) a.e.
- \(d(f,g)=\|f-g\|\) defines a metric on \(L^1(\mathbb{R}^d)\).
Theorem 2.4 (Riesz-Fischer) The vector space \(L^1\) is complete in its metric.
Corollary 2.4 If \(\{f_n\}_{n=1}^\infty\) converges to \(f\) in \(L^1\), the nthere exists a subsequence \(\{f_{n_k}\}_{k=1}^\infty\) such that \[ f_{n_k}(x)\to f(x)\quad a.e. x. \]
We say that a family \(\cal G\) of integrable functions is dense in \(L^1\) if for ant \(f\in L^1\) and \(\epsilon>0\), there exists \(g\in\cal G\) so that \(\|f-g\|<\epsilon\).
Theorem 2.5 The following families of functions are dense in \(L^1(\mathbb{R}^d)\):
- The simple functions.
- The step functions.
- The continuous functions of compact support.