Lebesgue Integration

Show that, in the above Dirichlet function example, $\left\{ f_{n}\right\}$ fails to converge to $f$ uniformly on $[0,1]$.

A partition $P^{\prime }$ of $[a, b]$ is called a refinement of a partition $P$ provided that each partition point of $P$ is also a partition point of $P^{\prime }$. For a bounded function $f:[a, b] \rightarrow \mathbb {R}$, show that under refinement lower Darboux sums do not decrease and upper Darboux sums do not increase.

Show that for a bounded function $f:[a, b] \rightarrow \mathbb {R}$, the left-hand side of (1) is no greater than the right-hand side.

Assume that $f:[a, b] \rightarrow \mathbb {R}$ is bounded. Show that $f:[a, b] \rightarrow \mathbb {R}$ is Riemann integrable if and only if there is a sequence $\left\{ P_{n}\right\}$ of partitions of $[a, b]$, each $P_{n+1}$ being a refinement of $P_{n}$, for which $\lim_{n \rightarrow \infty }\left[U\left(f, P_{n}\right)-L\left(f, P_{n}\right)\right] = 0$.

Use the preceding problem to show that a continuous function $f:[a, b] \rightarrow \mathbb {R}$, being uniformly continuous, is Riemann integrable.

Let $f:[0,1] \rightarrow \mathbb {R}$ be increasing. For a natural number $n$, define $P_{n}$ to be the partition of $[0,1]$ into $n$ subintervals of length $1 / n$. Show that $U\left(f, P_{n}\right)-L\left(f, P_{n}\right) \leq 1 / n[f(1)-f(0)]$. Use Problem 4 to show that $f:[0,1] \rightarrow \mathbb {R}$ is Riemann integrable.

Let $\left\{ f_{n}:[a, b] \rightarrow \mathbb {R}\right\}$ be a uniformly bounded sequence of Riemann integrable functions that converges uniformly to $f:[a, b] \rightarrow \mathbb {R}$. Show that $f:[a, b] \rightarrow \mathbb {R}$ also is Riemann integrable, and

Assume that $m(E) < \infty$ and let $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ be a sequence of bounded, measurable functions. Without referring to the Bounded Convergence Theorem or Egoroff's Theorem, give a direct proof that if $\left\{ f_{n}\right\} \rightarrow f$ uniformly on $E$, then $\lim_{n \rightarrow \infty } \int_{E} f_{n} d m = \int_{E} f d m$.

Let $m(E) < \infty$ and $\left\{ E_{n}\right\}_{n = 1}^{\infty }$ be a measurable partition of $E$. Just using the linearity of integration and the Bounded Convergence Theorem, show that $m(E) = \sum_{n = 1}^{\infty } m\left(E_{n}\right)$.

Let $f: E \rightarrow \mathbb {R}$ be bounded, finitely supported and measurable. For a measurable subset $A$ of $E$, show that $\left.\int_{A} f\right|_{A} d m = \int_{E} f \cdot \chi_{A} d m$.

Does the Bounded Convergence Theorem hold for the Riemann integral?

Let $f: E \rightarrow \mathbb {R}$ and $g: E \rightarrow \mathbb {R}$ be bounded, finitely supported and measurable, and assume that $f = g$ almost everywhere on $E$. Show that $\int_{E} f d m = \int_{E} g d m$.

Prove Theorem 2.

Theorem 2: If $f: \left[a, b\right] \rightarrow \mathbb {R}$ is a bounded, Riemann integrable function, then it is Lebesgue integrable and the two integrals are equal.

Let $\left\{ E_{n}\right\}_{n = 1}^{\infty }$ be a measurable partition of $E$. Just using the Monotone Convergence Theorem and the linearity of integration, show that $m(E) = \sum_{n = 1}^{\infty } m\left(E_{n}\right)$.

If $f:[a, b] \rightarrow [0, \infty ]$ is measurable, show that $\int_{[a, b]} f d m = \int_{(a, b)} f d m$.

Let $\left\{ u_{n}: E \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions. Define $s = \sum_{n = 1}^{\infty } u_{n}$. Show that $\int_{E} s d m = \sum_{n = 1}^{\infty } \int_{E} u_{n} d m$.

If $f:[a, b] \rightarrow [0, \infty ]$ is measurable, show that $\int_{[a, b]} f d m = 0$ if and only if $f = 0$ almost everywhere

Prove the linearity of integration for general coefficients $\alpha$ and $\beta$.

Let $\left\{ a_{n}\right\}$ be a sequence of non-negative real numbers. Define the function $f$ on $E = [1, \infty )$ by setting $f(x) = a_{n}$ if $n \leq x < n+1$. Show that $\int_{E} f d m = \sum_{n = 1}^{\infty } a_{n}$.

Let $f: E \rightarrow [0, \infty ]$ be measurable. Show that

where the supremum is taken over finitely supported, simple functions.

Let $\left\{ f_{n}: E \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions that converges pointwise on $E$ to $f$. Suppose $f_{n} \leq f$ on $E$ for each $n$. Show that $\lim_{n \rightarrow \infty } \int_{E} f_{n} d m = \int_{E} f d m$.

Show that the Monotone Convergence Theorem does not hold for decreasing sequences of functions.

Prove the following generalization of Fatou's Lemma: If $\left\{ f_{n}: E \rightarrow [0, \infty ]\right\}$ is a sequence of measurable functions, then $\int_{E}\left[\liminf f_{n}\right] d m \leq \liminf \int_{E} f_{n} d m$.

Prove the linearity of integration for general coefficients $\alpha$ and $\beta$.

Provide an example of a function $f:[0,1] \rightarrow \mathbb {R}$ that is not integrable, while $\left|f\right|:[0,1] \rightarrow \mathbb {R}$ is integrable.

Provide an example of an integrable function $f:[0,1] \rightarrow [0, \infty ]$ for which $f^{2} f r m[o]--:[0,1] \rightarrow [0, \infty ]$ is not integrable.

Let $f: E \rightarrow \mathbb {R}$ and $g: E \rightarrow \mathbb {R}$ be measurable and $f^{2}: E \rightarrow \mathbb {R}$ and $g^{2}: E \rightarrow \mathbb {R}$ be integrable. Show that $f \cdot g: E \rightarrow \mathbb {R}$ is integrable.

Let $f: E \rightarrow \mathbb {R}$ be integrable and $g: E \rightarrow \mathbb {R}$ be bounded and measurable. Show that $f \cdot g: E \rightarrow \mathbb {R}$ is integrable.

Let $f: E \rightarrow \mathbb {R}$ be integrable and $C$ a measurable subset of $E$. Show that the restriction of $f$ to $C$ is integrable and $\left.\int_{C} f\right|_{C} d m = \int_{E} f \cdot \chi_{C} d m$.

For a measurable function $f:[1, \infty ) \rightarrow \mathbb {R}$ that is bounded on bounded sets, for each $n$, define $a_{n} = \int_{[n, n+1)} f$. Is it true that $f$ is integrable if and only if the series $\sum_{n = 1}^{\infty } a_{n}$ converges? Is it true that $f$ is integrable if and only if the series $\sum_{n = 1}^{\infty } a_{n}$ converges absolutely?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable. Show that there is a sequence $\left\{ \psi_{n}: \mathbb {R} \rightarrow \mathbb {R}\right\}$ of finitely supported, simple functions that converges pointwise to $f$ and

Prove the General Dominated Convergence Theorem by following the proof of the Dominated Convergence Theorem, but replacing the sequences $\left\{ g-f_{n}\right\}$ and $\left\{ g+f_{n}\right\}$, respectively, by $\left\{ g_{n}-f_{n}\right\}$ and $\left\{ g_{n}+f_{n}\right\}$.

Let $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ be a sequence of integrable functions for which $f_{n} \rightarrow f$ pointwise on $E$ and $f$ is integrable over $E$. Show that $\int_{E}\left|f-f_{n}\right| d m \rightarrow 0$ if and only if $\lim_{n \rightarrow \infty } \int_{E}\left|f_{n}\right| d m = \int_{E}\left|f\right| d m$.

Let $f$ be a real-valued function of two variables $(x, y)$ that is defined on the square $Q = \left\{ (x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\right\}$ and is a measurable function of $x$ for each fixed value of $y$. Suppose for each fixed value of $x, \lim_{y \rightarrow 0} f(x, y) = f(x)$ and that for all $y$, we have $\left|f(x, y)\right| \leq g(x)$, where $g$ is integrable over $[0,1]$. Show that

Also show that if the function $f(x, y)$ is continuous in $y$ for each $x$, then

is a continuous function of $y$.

Let $f$ be a real-valued function of two variables ($x, y$) that is defined on the square $Q = \left\{ (x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\right\}$ and is a measurable function of $x$ for each fixed value of $y$. For each $(x, y) \in Q$, let the partial derivative $\partial f / \partial y$ exist. Suppose there is a function $g$ that is integrable over $[0,1]$ and such that

Prove that

Prove the theorem regarding the continuity of integration.

Let $f:[0, \infty ] \rightarrow \mathbb {R}$ be integrable. Show that

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable. Show that the function $F: \mathbb {R} \rightarrow \mathbb {R}$ defined by

is properly defined and continuous. Is it Lipschitz?

For each of the two functions $f$ on $[1, \infty )$ defined below, show that $\lim_{n \rightarrow \infty } \int_{[1, n]} f d m$ exists while $f$ is not integrable over $[1, \infty )$. Does this contradict the continuity of integration?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable. Show that the following four assertions are equivalent:

Let $f:[a, b] \rightarrow \mathbb {R}$ be integrable and $\epsilon > 0$. Establish the following approximations.