1. Lebesgue Measure and Integration
2. Properties of Outer Measure and Measurable Sets
3. Lemmas 7.4.2 and 7.4.4
4. Lemmas 7.4.6 - 7.4.11
5. Measurable Function, Regularity
6. Product and Slices
7. Pugh Lebesgue Integral 1
8. Pugh Lebesgue Integral 2
9. Tao Lebesgue Integral 1
10. Tao Lebesgue Integral 2

HW1 (Updated 1/26)
HW2 (Updated 2/3)
HW3 (Updated 2/11)
HW4 (Updated 2/17)
HW5 (Updated 2/25)
HW6 (Updated 3/2)
HW7 (Updated 3/11)
HW8 (Updated 3/18)
HW9 (Updated 3/25)
HW10 (Updated 4/6)
HW11 (Updated 4/16)
HW12 (Updated 4/23)

The Lebesgue integral is defined in terms of the undergraph. For a function $f: \mathbb{R}\rightarrow [0,\infty)$, the undergraph of $f$ is defined as $$u(f)=\{(x,y):0\leq y <f(x)\}$$ If $u(f)$ is measurable, we say that $f$ is measurable, and define the Lebesgue integral of $f$ as $$\int f = m(u(f))$$ If $\int f < \infty$, we say that $f$ is Lebesgue integrable. From this definition, we notice that the Lebesgue integral heavily depends on the notion of measure and measurability; this mode of integration has many benefits compared to the Riemann integral.

The Riemann integral is defined in terms of upper and lower integrals. For a function $f: [a,b] \rightarrow \mathbb{R}$, the upper and lower sums are defined as $$U(f,P)=\sum_{k=1}^{n}M_k (f)(x_k - x_{k-1})$$ $$L(f,P)=\sum_{k=1}^{n}m_k (f)(x_k - x_{k-1})$$ where $P$ is a partition of hte set $[a,b]$ and $$M_k (f) = sup \{f(x):x \in [x_{k-1},x_k] \}$$ $$m_k (f) = inf \{f(x):x \in [x_{k-1},x_k] \}$$ Then, the upper and lower integrals of $f$ are defined as $$U(f)=inf\{U(f,P): \text{P is a partition of [a,b]}\}$$ $$L(f)=sup\{L(f,P): \text{P is a partition of [a,b]}\}$$ Lastly, the Riemann integral over $[a,b]$ is defined as $$\int_{a}^{b}f(x)dx = L(f)=U(f)$$ From this definition, we notice that the notion of Riemann integral heavily depends on the interval on which it is defined. This could generate concerns over the flexibility of the Riemann integral that Lebesgue integral can handle more elegantly. For example, when both types of integrals are generalized to higher dimensions, the notion of an interval in Riemann integral is more difficult to define since we can have “irregular shapes” such as circles. On the other hand, Lebesgue integral can compute the measure of the undergraph defined on these “irregular shapes” more easily using box covers. I do think, however, measure is a rather abstract concept, its computation is a bit more difficult to understand compared to the “area under the curve” definition of the Riemann integral.

We start with definition of Brownian motion.

Let $\omega : [0, \infty) \rightarrow \mathbb{R}^{n}$ be a path. Given $t_1 < t_2$ and $\omega(t_1) = x_1$, the probability density for $\omega(t_2)$ is
$$p(t_2-t_1, x-x_1) = \frac{1}{[4\pi (t_2-t_1) ]^{\frac{n}{2}}} e^{\frac{-|x-x_1|^2}{4(t_2-t_1)}}$$ In addition, for any $t_1\leq t \leq t_2$, $\omega(t)$ does not depend on the trajectory of the path before $t_1$. The definition of Brownian motion provides a probability integral that motivates the construction of the measure on the space of Brownian motion paths. Namely, give $0\leq t_1 \leq t_2 \leq \cdot \cdot \cdot \leq t_k$ and Borel sets $E_j \subset \mathbb{R}^{n}$, and starting at $x=0$ and $t=0$, we have $$Pr[\omega(t_1) \in E_1, \omega(t_2) \in E_2, \cdot \cdot \cdot,\omega(t_k) \in E_k] = \int_{E_1} \cdot \cdot \cdot \int_{E_k} p(t_k - t_{k-1}, x_k - x_{k-1}) \cdot \cdot \cdot p(t_1, x_1) dx_k \cdot \cdot \cdot dx_1$$ One last thing before we construct the measure is we have to define the space of Brownian motion paths. We characterize the path by its location at positive rational time t, and the space of all paths is $$\mathcal{P} = \prod_{t \in \mathbb{Q}^{+}} \dot{\mathbb{R}}^{n}$$ where $\dot{\mathbb{R}}^{n} = \mathbb{R}^{n} \cup \{ \infty \}$ is the one-point compactification of $\mathbb{R}^{n}$. Hausdorff's theorem (Wright 1994) ensures that $\mathcal{P}$ is compact. Now, we are ready to construct our measure, motivated by the following theorem:

If $X$ is a compact metric space and $\alpha$ is a positive linear functional on $C(X)$, then there exists a unique finite, positive Borel measure $\mu$ such that $$\alpha(f) = \int f d\mu$$ for all $f \in C(X)$

proof: see Taylor (2006) Theorem 13.5
To use this theorem, we construct a positive linear functional $E: C(\mathcal{P}) \rightarrow \mathbb{R}$. We first define $E$ on the subspace $C^{\#}$ with only continuous functions that depend on only finitely many of the factors in $\mathcal{P}$, with the form $$\phi (\omega) = F(\omega(t_1), … , \omega(t_k)) \text{ , } t_1 < … <t_k$$ where $F$ is continuous on $\mathcal{P}$ and $t_j \in \mathbb{Q}^{+}$. Then we let $$E(\phi) = \int \cdot \cdot \cdot \int p(t_1, x_1) p(t_2 - t_1, x_2 - x_1) \cdot \cdot \cdot p(t_k - t_{k-1}m x_k - x_{k-1}) F(x_1, …, x_k) dx_k \cdot \cdot \cdot dx_1$$ To check this functional is well-defined, we resort to the following proposition:

For $t \neq s$, $\int p(t, x-y)p(s,y)dy = p(t+s, x)$
proof: see Sternberg (2014) slides 10 and 11
Proposition 1 showed that if $F$ does not depend on a given set of $x_i$, we obtain the same functional $E(\phi)$. In addition, we have $E(1) = 1$, so by the Stone-Weierstrass Theorem, $C^{\#}$ is dense in $C(\mathcal{P})$ (see Taylor 2006 Theorem A.23). Furthermore, this functional can be extended to $C(\mathcal{P})$. Therefore, by Theorem 1, we have a measure on the space $\mathcal{P}$:

There exists a unique Borel measure (called the Wiener Measure) $\mu$ on $\mathcal{P}$ such that $$ E(\phi) = \int_{\mathcal{P}} \phi(\omega) d \mu(\omega)$$ for each $\phi(\omega)$ with a continuous $F$ on $\mathcal{P}$

Sternberg, Shlomo Z. 2014. “Wiener Measure.” Harvard Math 201a, November 11.

Taylor, Michael E. 2006. Measure Theory and Integration. Graduate Studies in Mathematics, v. 76. Providence, R.I: American Mathematical Society.

Wright, David G. 1994. “Tychonoff’s Theorem.” Proceedings of the American Mathematical Society 120 (3): 985–87. https://doi.org/10.1090/S0002-9939-1994-1170549-2.

A paper that constructs a set which includes points at which the density of the set can take on any values in $[0,1]$