Analysis Theorems

10 Integration

Lemma 10.1. A countable union \(\cup _iU_i\) of measure \(0\) sets is measure \(0\).

Proof. For any \(\ee >0\), cover each \(U_i\) with countably many cells summing to size \(\leq \frac {\ee }{2^{i+1}}\). □

Theorem 10.2 (Riemann-Lebesgue Theorem). A bounded function \(f\) on a closed cell \(\Delta \) is Riemann integrable (\(\int _{*\Delta }f=\int ^{*\Delta }f\)) iff \(f\) is continuous almost everywhere.

Proof. Let \(M\) be the bound for \(f\). The set of points with oscillation \(\leq \ee \) is open, so the set of discontinuities is compact, and since \(f\) is discontinuous on a set of measure \(0\), by compactness this is actually content \(0\).

If \(f\) is continuous almost everywhere, for any \(\ee >0\), choose a partition \(\cP \) such that \(f\) varies by at most \(\frac {\ee }{2|\Delta |}\) in each cell where \(f\) is continuous, and so that the the discontinuous points are covered by cells with total content less than \(\frac {\ee }{2M|\Delta |}\). Then we have \(S^*(f,\cP )-S_*(f,\cP ) = \sum _{\Delta \in \cP }o_f(\Delta )|\Delta | < \frac {\ee }{2} + \frac {\ee }{2} = \ee \) so \(\int _{*\Delta }f=\int ^{*\Delta }f\).

Conversely if \(f\) is not continuous almost everywhere, by Lemma 10.1 there is an \(\ee \) such that the set of points with oscillation \(\geq \ee \) cannot be covered by cells of size smaller than \(\delta \), so \(\sum _{\Delta \in \cP }o_f(\Delta )|\Delta |\) is larger than than \(\ee \delta \). □

Note for some of the following theorems the conditions on the function may be made weaker, ie. it can be just continuous, differentiable on the interior, with a bounded and almost everywhere continuous derivative.

Theorem 10.3 (Second Fundamental Theorem of Calculus). If \(f:\RR \to \RR \) is \(\cC ^1\) near \([a,b]\), then \(\int _{[a,b]}f' = f(b)-f(a)\).

Proof. For any partition \(\cP \) we have by Proposition 4.2 that for each \([x,y] = \Delta \in \cP \), \(f'(c)(y-x) = f(y)-f(x)\) for a \(c \in \Delta \), and as this is true for any partition, \(\int _*f'\leq f(b)-f(a) \leq \int ^*f'\), so by Theorem 10.2 we are done. □

Corollary 10.4 (Integration by Parts). If \(f,g\) are \(\cC ^1\) near an interval \([a,b]\), then \(\int _{[a,b]}fg' = f(b)g(b)-f(a)g(a) - \int _{[a,b]}f'g\)

Proof. This follows from linearity, Corollary 4.4, and Theorem 10.3. □

Theorem 10.5 (First Fundamental Theorem of Calculus). If \(f:\RR \to \RR \) is continuous on the interval \([a,b]\), then \(F(x) = \int _a^xf\) is continuous on the interval and \(F'(x) = f(x)\) in the interior.

Proof.

\[F(x+h)-F(x)-f(x)h=\int _x^{x+h}f(t)-f(x)dt\]

and as \(|f(t)-f(x)| \leq \ee \) for small enough \(h\) and any \(\ee >0\), we have

\[\int _x^{x+h}f(t)-f(x)dt \leq \int _x^{x+h}\ee \leq \ee h\]

□

Theorem 10.6 (Linearity of the Integral). If \(f,g\) are Riemann integrable and \(D\) is a Jordan domain, \(\int _D(c_1f+c_2g) = c_1\int _Df+c_2\int _Dg\)

Proof. Scaling is obvious, so it suffices to prove \(\int _D (f+g) = \int _D f + \int _D g\). To see this, choose partitions \(\cP _1\) and \(\cP _2\) so that \(S_*(f,\cP _1),S_*(g,\cP _2)\) differ by at most \(\frac {\ee }{2}\) from \(\int _D f,\int _D g\). Then taking a common refinement \(\cP \), we have that \(S_*(f+g)\) differs from \(\int _D (f+g)\) by at most \(\ee \). A similar argument can be made for the upper integral. □

Theorem 10.7 (Positivity of the Integral). If \(f\geq 0\) on \(D\), then \(\int _Df\geq 0\). If \(f\geq g\) on \(D\), then \(\int _Df\geq \int _Dg\). Also for any \(f\), \(|\int _Df|\leq \int _D|f|\).

Proof. The first is obvious by looking at any partition. The second follows from the first and Theorem 10.6. The last follows from the second by noting \(|f|\geq f,-f\). □

Theorem 10.8. For a vector valued function \(f\), \(\Vert \int _Df\Vert _2\leq \int _D\Vert f\Vert _2\).

Proof. Let \(u\) be a unit vector in the direction of \(\int _Df\). Then by the Cauchy-Schwarz inequality and linearity, we have

\[\bigg \Vert \int _Df\bigg \Vert _2=u\cdot \int _Df = \int _Du\cdot f \leq \int _D\Vert f\Vert _2\]

□

Theorem 10.9 (Invariance of the Integral). If \(f,g\) are integrable on a Jordan domain \(D\) and differ on a set of content \(0\), then \(\int _Df = \int _Dg\).

Proof. It suffices to show a function nonzero on a set of content \(0\) has integral \(0\), but this is true by definition of content \(0\), and the fact that the function must be bounded. □

Theorem 10.10 (Additivity of the Integral). If \(f\) is integrable on \(D,E\), Jordan domains whose intersection is content \(0\), then \(\int _{D\cup E}f = \int _{D}f + \int _{E}f\).

Proof. If \(\chi \) denotes the characteristic function, then \(\int _{D\cup E}f = \int _{D\cup E}(\chi _Df +\chi _Ef) = \int _Df + \int _Ef\), where we have ignored the boundary as it is content \(0\). □

Theorem 10.11 (Fubini’s Theorem). If \(f\) is integrable in the product cell \(\Delta = \Delta _I\times \Delta _{II}\), and the functions \(\int _{*\Delta _I}f,\int ^*_{\Delta _I}f\) are integrable, then \(\int _\Delta f = \int _{\Delta _{II}}\int _{*\Delta _I}f = \int _{\Delta _{II}}\int ^*_{\Delta _I}f\).

Proof. Suppose we have a partition \(\cP \) that is the product of the partitions \(\cP _I,\cP _{II}\). Then we have

\[\sum _{\Delta \in \cP }\inf _{x \in \Delta }f(x)|\Delta |=\sum _{\Delta _{II} \in \cP _{II}}\sum _{\Delta _I \in \cP _I}\inf _{x\in \Delta _I\times \Delta _{II}}f(x)|\Delta _I| |\Delta _{II}|\]

\[\leq \sum _{\Delta _{II} \in \cP _{II}}\inf _{x_{II}\in \Delta _{II}}\sum _{\Delta _I \in \cP _I}\inf _{x_{I} \in \Delta _{I}}f(x_I,x_{II})|\Delta _I||\Delta _{II}| \]

\[\leq \sum _{\Delta _{II} \in \cP _{II}}\inf _{x_{II}\in \Delta _{II}}\int _{*\Delta _{I}}f(x_I,x_{II})|\Delta _{II}|\]

\[\leq \sum _{\Delta _{II} \in \cP _{II}}\sum _{\Delta _I\in \cP _I}\sup _{x_{II}\in \Delta _{II}}\sup _{x_I\in \Delta _{I}}f(x_I,x_{II})|\Delta _I||\Delta _{II}| \leq \sum _{\Delta \in \cP }\sup _{x \in \Delta }f(x)|\Delta | \]

showing that \(\int _{*\Delta } f \leq \int _{\Delta _{II}}\int _{*\Delta _I}f\leq \int ^*_{\Delta }f\) but as \(f\) is integrable these are equalities. The other equality comes from dualizing the argument. □

Theorem 10.12. If \(f_n\) is a uniformly convergent sequence of integrable functions in a cell \(\Delta \), the limit \(f\) is integrable and \(\lim _{n\to \infty }\int _\Delta f_n=\int _\Delta f\).

Proof. To see \(f\) is integrable, note that the union of the discontinuities of the \(f_n\) is measure \(0\) by Lemma 10.1, so by Theorem 3.16 \(f\) is continuous away from these points. \(|\int _{\Delta } f-\sum _1^nf_n|\leq \int _{\Delta }| f-\sum _1^nf_n| \to 0\) as \(n \to \infty \), so this concludes the proof. □

Proposition 10.13. Any open set \(U \subset \RR ^m\) is the union of countably many open Jordan domains \(D_i\) with \(\bar D_i \subset D_{i+1}\) and the \(D_i\) composed of interiors of unions of cells of a partition.

Proof. Intersect \(U\) with a ball radius \(n\), partition \(\RR ^m\) into cells of size \(\frac {1}{2^n}\) and take the interior of the union of the pieces whose boundary is completely contained inside \(U\) as \(D_n\). To make sure \(\bar {D_n}\subset D_{n+1}\), note that the boundary of \(D_n\) is compact, so is eventually covered by another \(D_i\), so we can take a nice enough subsequence. □

Lemma 10.14. If \(k<n\) and \(V\) is a \(\cC ^1\) \(k\)-submanifold of \(\RR ^n\), then \(V\) is measure \(0\).

Proof. By Lemma 10.1 it suffices to do this locally, and by Corollary 9.5 \(V\) is locally given by implicit \(\cC ^1\) functions \(g_i\), and so locally the derivatives are uniformly continuous, and so for any we can cover \(V\) in the plane of the variables that the \(g_i\) are a function of finitely many by cells of height arbitrarily small. □

Theorem 10.15 (Change of Variables). If \(D,E\) are open sets of \(\RR ^n\), and \(\phi :D\to E\) is a \(\cC ^1\) diffeomorphism, then if \(f\) has a finite improper integral on \(E\), then \((f\circ \phi )|\det \phi '|\) has a finite improper integral on \(D\), moreover \(\int _D^\smile (f\circ \phi )|\det \phi '| = \int _E^\smile f\).

Proof. We will make a series of reductions of this problem. If \(E_i\) is a sequence of Jordan domains covering \(E\) of the sort in Proposition 10.13 with \(\bar {E_i}\subset E_{i+1}\), then \(\phi ^{-1}(E_i)\) is a sequence of Jordan domains (the boundaries are submanifolds by Corollary 9.7 hence are measure \(0\) by Lemma 10.14) with \(\phi ^{-1}(\bar {E}_i) = \overline {\phi ^{-1}(E_i)}\subset \phi ^{-1}(E_{i+1})\) covering \(E\), so it suffices to show this where \(f\) is a positive function in a cell \(\Delta \) in \(E\). By passing to refinements (Theorem 10.10) and applying Theorem 9.2, Lemma 3.14, and the chain rule, it suffices to prove this when \(\phi \) changes only \(1\) variable on a sufficiently small cell \(\Delta \).

Finally to reduce to when \(f\) is a constant on the cell, suppose this has been proven, and for any partition \(\cP \) of the cell \(\Delta \), define \(f^*\) as \(f\) on the boundary of each \(\Delta ' \in \cP \) and \(\sup _{\Delta '}f\) on the inside. Then we have:

\[S^*(f,\cP ) = \int _\Delta f^* = \int _{\phi ^{-1}\Delta }(f^*\circ \phi )|\det \phi '| \geq \int _{\phi ^{-1}\Delta }(f\circ \phi )|\det \phi '| \]

so \(\int ^*_{\Delta }f\geq \int _{\phi ^{-1}\Delta }(f\circ \phi )|\det \phi '|\). Dually we get \(\int _{*\Delta }f\leq \int _{\phi ^{-1}\Delta }(f\circ \phi )|\det \phi '|\).

Now we consider the case when \(f\) is constant (even \(f=1\) suffices) and will reduce to the \(1\)-dimensional case. We label the coordinates in \(\phi ^{-1}(\Delta )\) \(x_1\) and \(x_{II}\) and in \(\Delta \) \(y_1\) and \(y_{II}\) to distinguish the coordinate \(\phi \) changes. \(\phi '\) looks like

\[\begin {pmatrix} \partial _1\phi _1&\partial _{II}\phi _1\\ 0&I_{n-1} \end {pmatrix}\]

so \(\det \phi ' = \partial _1\phi _1\). Now from the \(1\)-dimensional case and Theorem 10.11, we get

\[\int _\Delta 1 = \int _{y_{II} \in \Delta _{II}}\int _{y_1\in \Delta _1}1 = \int _{x_{II} \in \Delta _{II}}\int _{x_1 \in \phi ^{-1}\Delta _1}|\partial _1\phi _1|\]

\[=\int _{x_{II}\in \Delta _{II}}\int _{x_1\in \phi ^{-1}\Delta _1}|\phi '| = \int _{\phi ^{-1}\Delta }|\phi '|\]

Finally for the case of \(1\) dimension, if \(\phi :[a,b]\to [c,d]\) is our function, then \(\int _{[c,d]}f = F(d)-F(c)\) by Theorem 10.5, and since \((F\circ \phi )' = (f\circ \phi )\phi '\) and WMA \(\phi '>0\) as \(\phi \) is a \(\cC ^1\) diffeomorphism (one treats the \(\phi '<0\) case similarly), we have \(\int _{[a,b]}(f\circ \phi )|\phi '| = \int _{[a,b]}(F\circ \phi )'\) = \(F(d)-F(c)\). □