Integration

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f: X \rightarrow [0, \infty ]$ is an $\mathcal{S}$-measurable function such that $\int f d \mu < \infty$. Explain why

for each set $E \in \mathcal{S}$ with $\mu (E) = \infty$.

Suppose $X$ is a set, $\mathcal{S}$ is a $\sigma$-algebra on $X$, and $c \in X$. Define the Dirac measure $\delta_{c}$ on $(X, \mathcal{S})$ by

Prove that if $f: X \rightarrow [0, \infty ]$ is $\mathcal{S}$-measurable, then $\int f d \delta_{c} = f(c)$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f: X \rightarrow [0, \infty ]$ is an $\mathcal{S}$-measurable function. Prove that

Give an example of a Borel measurable function $f:[0,1] \rightarrow (0, \infty )$ such that $L(f,[0,1]) = 0$.

Verify the assertion that integration with respect to counting measure is summation (Example 3.6).

Suppose $(X, \mathcal{S}, \mu )$ is a measure space, $f: X \rightarrow [0, \infty ]$ is $\mathcal{S}$-measurable, and $P$ and $P^{\prime }$ are $\mathcal{S}$-partitions of $X$ such that each set in $P^{\prime }$ is contained in some set in $P$. Prove that $\mathcal{L}(f, P) \leq \mathcal{L}\left(f, P^{\prime }\right)$.

Suppose $X$ is a set, $\mathcal{S}$ is the $\sigma$-algebra of all subsets of $X$, and $w: X \rightarrow [0, \infty ]$ is a function. Define a measure $\mu$ on $(X, \mathcal{S})$ by

for $E \subseteq X$. Prove that if $f: X \rightarrow [0, \infty ]$ is a function, then

where the infinite sums above are defined as the supremum of all sums over finite subsets of $E$ (first sum) or $X$ (second sum).

Suppose $\lambda$ denotes Lebesgue measure on $\mathbb {R}$. Give an example of a sequence $f_{1}, f_{2}, \ldots$ of simple Borel measurable functions from $\mathbb {R}$ to $[0, \infty )$ such that $\lim_{k \rightarrow \infty } f_{k}(x) = 0$ for every $x \in \mathbb {R}$ but $\lim_{k \rightarrow \infty } \int f_{k} d \lambda = 1$.

Suppose $\mu$ is a measure on a measurable space $(X, \mathcal{S})$ and $f: X \rightarrow [0, \infty ]$ is an $\mathcal{S}$-measurable function. Define $\nu : \mathcal{S} \rightarrow [0, \infty ]$ by

for $A \in \mathcal{S}$. Prove that $\nu$ is a measure on $(X, \mathcal{S})$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f_{1}, f_{2}, \ldots$ is a sequence of nonnegative $\mathcal{S}$-measurable functions. Define $f: X \rightarrow [0, \infty ]$ by $f(x) = \sum_{k = 1}^{\infty } f_{k}(x)$. Prove that

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f_{1}, f_{2}, \ldots$ are $\mathcal{S}$-measurable functions from $X$ to $\mathbb {R}$ such that $\sum_{k = 1}^{\infty } \int \left|f_{k}\right| d \mu < \infty$. Prove that there exists $E \in \mathcal{S}$ such that $\mu (X \backslash E) = 0$ and $\lim_{k \rightarrow \infty } f_{k}(x) = 0$ for every $x \in E$.

Show that there exists a Borel measurable function $f: \mathbb {R} \rightarrow (0, \infty )$ such that $\int \chi_{I} f d \lambda = \infty$ for every nonempty open interval $I \subseteq \mathbb {R}$, where $\lambda$ denotes Lebesgue measure on $\mathbb {R}$.

Give an example to show that the Monotone Convergence Theorem (3.11) can fail if the hypothesis that $f_{1}, f_{2}, \ldots$ are nonnegative functions is dropped.

Give an example to show that the Monotone Convergence Theorem can fail if the hypothesis of an increasing sequence of functions is replaced by a hypothesis of a decreasing sequence of functions.

Suppose $\lambda$ is Lebesgue measure on $\mathbb {R}$ and $f: \mathbb {R} \rightarrow [-\infty , \infty ]$ is a Borel measurable function such that $\int f d \lambda$ is defined.

Suppose $\mathcal{S}$ and $\mathcal{T}$ are $\sigma$-algebras on a set $X$ and $\mathcal{S} \subseteq \mathcal{T}$. Suppose $\mu_{1}$ is a measure on $(X, \mathcal{S}), \mu_{2}$ is a measure on $(X, \mathcal{T})$, and $\mu_{1}(E) = \mu_{2}(E)$ for all $E \in \mathcal{S}$. Prove that if $f: X \rightarrow [0, \infty ]$ is $\mathcal{S}$-measurable, then $\int f d \mu_{1} = \int f d \mu_{2}$.

Suppose that $(X, \mathcal{S}, \mu )$ is a measure space and $f_{1}, f_{2}, \ldots$ is a sequence of nonnegative $\mathcal{S}$-measurable functions on $X$. Define a function $f: X \rightarrow [0, \infty ]$ by $f(x) = \liminf_{k \rightarrow \infty } f_{k}(x)$.

Give an example of a sequence $x_{1}, x_{2}, \ldots$ of real numbers such that

but $\int x d \mu$ is not defined, where $\mu$ is counting measure on $\mathbb {N}$ and $x$ is the function from $\mathbb {N}$ to $\mathbb {R}$ defined by $x(k) = x_{k}$.

Show that if $(X, \mathcal{S}, \mu )$ is a measure space and $f: X \rightarrow [0, \infty )$ is $\mathcal{S}$-measurable, then

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f_{1}, f_{2}, \ldots$ is a monotone (meaning either increasing or decreasing) sequence of $\mathcal{S}$-measurable functions. Define $f: X \rightarrow [-\infty , \infty ]$ by

Prove that if $\int \left|f_{1}\right| d \mu < \infty$, then

Henri Lebesgue wrote the following about his method of integration: I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

Use 3.15 to explain what Lebesgue meant and to explain why integration of a function with respect to a measure can be thought of as partitioning the range of the function, in contrast to Riemann integration, which depends upon partitioning the domain of the function.

Give an example of a sequence $f_{1}, f_{2}, \ldots$ of functions from $\mathbb {N}$ to $[0, \infty )$ such that

for every $m \in \mathbb {N}$ but $\lim_{k \rightarrow \infty } \int f_{k} d \mu = 1$, where $\mu$ is counting measure on $\mathbb {N}$.

Give an example of a sequence $f_{1}, f_{2}, \ldots$ of continuous functions from $\mathbb {R}$ to $[0,1]$ such that

for every $x \in \mathbb {R}$ but $\lim_{k \rightarrow \infty } \int f_{k} d \lambda = \infty$, where $\lambda$ is Lebesgue measure on $\mathbb {R}$.

Suppose $\lambda$ is Lebesgue measure on $\mathbb {R}$ and $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Borel measurable function such that $\int \left|f\right| d \lambda < \infty$. Define $g: \mathbb {R} \rightarrow \mathbb {R}$ by

Prove that $g$ is uniformly continuous on $\mathbb {R}$.

Let $\lambda$ denote Lebesgue measure on $\mathbb {R}$. Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a Borel measurable function such that $\int \left|f\right| d \lambda < \infty$. Prove that

Let $\lambda$ denote Lebesgue measure on $\mathbb {R}$. Give an example of a continuous function $f:[0, \infty ) \rightarrow \mathbb {R}$ such that $\lim_{t \rightarrow \infty } \int_{[0, t]} f d \lambda$ exists (in $\mathbb {R}$) but $\int_{[0, \infty )} f d \lambda$ is not defined.

Let $\lambda$ denote Lebesgue measure on $\mathbb {R}$. Give an example of a continuous function $f:(0,1) \rightarrow \mathbb {R}$ such that $\lim_{n \rightarrow \infty } \int_{\left(\frac{1}{n}, 1\right)} f d \lambda$ exists (in $\mathbb {R}$) but $\int_{(0,1)} f d \lambda$ is not defined.

Verify the assertion in 3.38.

Verify the assertion in Example 3.41.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $f \in \mathcal{L}^{1}(\mu )$. Prove that

is the countable union of sets with finite $\mu$-measure.

Suppose

Prove that $\lim_{k \rightarrow \infty } \int_{0}^{1} f_{k} = 0$.

Give an example of a sequence of nonnegative Borel measurable functions $f_{1}, f_{2}, \ldots$ on $[0,1]$ such that both the following conditions hold:

• $\lim_{k \rightarrow \infty } \int_{0}^{1} f_{k} = 0$;

• $\sup_{k \geq m} f_{k}(x) = \infty$ for every $m \in \mathbb {N}$ and every $x \in [0,1]$.

Let $\lambda$ denote Lebesgue measure on $\mathbb {R}$.

Prove or give a counterexample: If $G$ is an open subset of $(0,1)$, then $\chi_{G}$ is Riemann integrable on $[0,1]$.

Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$.