Integral

Prove or disprove that the composition of any two Lebesgue integrable functions with compact support $f, g : \mathbb {R} \to \mathbb {R}$ is still integrable.

Let $f \in L^1(0,1)$. Assume that for any $x \in (0,1)$ and every $\varepsilon > 0$, there is an open interval $J_x \subset (0,1)$ such that

Prove that for every open interval $I \subset (0,1)$

Let $(X, \mathcal{M}, \mu )$ be a positive measure space with $\mu (X) < \infty$. Show that a measurable function $f : X \to [0,\infty )$ is integrable (i.e., one has $\int_X f \, d\mu < \infty$) if and only if the series

converges.

Let $E$ be a Banach space, $(X, \pi , \mu )$ a probability space, and $f : X \to E$ such that $g \circ f$ is $\mu$-integrable for every $g \in E'$.

Define

Does $L \in E''$? Justify your answer.

Let $(X, \mathcal{M}, \mu )$ be a positive measure space with $\mu (X) < \infty$, and let $f$ and $g$ be real-valued measurable functions with

Show that either (a) $f = g$ a.e., or (b) there exists an $E \in \mathcal{M}$ such that

Let $(X, \mathcal{M}, \mu )$ be a positive measure space, and $S$ a closed set in $\mathbb {C}$. Suppose that

whenever $E \in \mathcal{M}$ and $\mu (E) > 0$. Show that $\{ x \in X \mid f(x) \notin S\}$ has measure $0$.

Let $f : [0,1] \to (0,\infty )$ and let $0 < \alpha < 1$. Show that

where $\inf$ is extended over all measurable $E \subset [0,1]$ with $m(E) \geq \alpha$.

Let $\{ f_n\}$ be a sequence of real-valued functions in $L^1(\mathbb {R})$ and suppose that for some $f \in L^1(\mathbb {R})$

Prove that $f_n \to f$ almost everywhere with respect to Lebesgue measure.

Let $\mu$ be a finite measure on $\mathbb {R}$, and define

Show that $f(x)$ is finite a.e.with respect to the Lebesgue measure on $\mathbb {R}$.

Let $(X, \mathcal{M}, \mu )$ be a positive measure space, $f_n : X \to [0,\infty ]$ a sequence of integrable functions, and $f : X \to [0,\infty ]$ an integrable function. Suppose that $f_n \to f$ a.e.$[\mu ]$, and that

Prove that

Let $\mu$ be a $\sigma$-finite, positive measure on a $\sigma$-algebra $\mathcal{M}$ in a set $X$.

Let $m$ denote the Lebesgue measure on $[0,1]$ and let $(f_n)$ be a sequence in $L^1(m)$ and $h$ a non-negative element of $L^1(m)$. Suppose that

Let $\{ f_n\}$ be a sequence of non-negative measurable functions in $L^p(\mathbb {R})$ for some $1 \leq p < \infty$. Show that $f_n \to f$ $(L^p)$ if and only if $f_n^p \to f^p$ $(L^1)$.

Let $1 \leq p < \infty$. All parts refer to Lebesgue measure on $\mathbb {R}$.

Suppose $f_n$ is a sequence of measurable functions on $[0,1]$ with $f_n \to 0$ a.e.on $[0,1]$ and $\int_0^1 f_n^2 \, dm \leq 1$ for all $n$. Prove

Let $(X, \mathcal{M}, \mu )$ be a probability space (a positive measure space with $\mu (X) = 1$). Show that if $f$ is an integrable function with values in $[1, \infty )$, then

Let $\{ a_n\}_{n \geq 2}$ be a sequence of real numbers with $\left|a_n\right| \leq \log n$. Consider

Let $S$ be a bounded Lebesgue measurable set in $\mathbb {R}$, and let $\{ c_n\}$ be a sequence in $\mathbb {R}$. Show that

where $\lambda$ is Lebesgue measure.

Let $(X, \mathcal{M}, \mu )$ be a measure space and $f \in L(\mu )$. Evaluate

Suppose that $f$ is a measurable real valued function on $\mathbb {R}$ such that $t \mapsto e^{tx} f(t)$ is in $L^1(\mathbb {R})$ for all $x \in (-1,1)$. Define the function

for all $x \in (-1,1)$. Prove that $\varphi$ is differentiable on $(-1,1)$.

Evaluate

justifying any interchange of limits you use.

Let $(X, \mathcal{A}, \mu )$ be a probability space and $f : X \to [1,\infty )$ a measurable function. Prove or disprove:

For $i = 1,2$, let $X_i = \mathbb {N}$, let $\mathcal{M}_i = 2^{\mathbb {N}}$ (the $\sigma$-algebras of all subsets of $\mathbb {N}$) and let $\mu_i$ be the counting measure. For the function $f : X_1 \times X_2 \to \mathbb {R}$ defined by

compute the iterated integrals

How do you reconcile your answers with Fubini's Theorem?

Let $(\Omega , \mu )$ be any measure space. For $f \in L^1(\Omega , \mu )$, and for $\lambda > 0$ define

and

Show that the functions $\varphi$ and $\psi$ are Borel measurable and that

Let $f \in L^2(0,1)$. Set

Show that

for each $h$, $0 < h < 1$, where $c$ is a positive number independent of $f$.

Let $f, g \in L^1([0,1])$ and assume that $f(x)g(y) = f(y)g(x)$ for all $x, y \in [0,1]$. Show that

where

and $dA$ denotes the planar Lebesgue measure.

Give an example of a Baire-1 function which is not Riemann integrable. Explain why such a function would have to be Lebesgue integrable.

Let $(\Omega , \mathcal{A}, \mu )$ be a $\sigma$-finite measure space, $f : \Omega \to \mathbb {R}$ measurable. Suppose there is a $c \in \mathbb {R}$ such that for all $X \subset \Omega$ of finite measure $\left|\int_X f \, d\mu \right| \leq c$ holds. Prove that $f \in L^1(\Omega , \mathcal{A}, \mu )$.

Let $(X, \mathcal{M}, \mu )$ be a measure space. Two functions $f, g : X \to \mathbb {R}$ are said to be comonotone if

for every $(x,y) \in X \times X$. If $\mu (X) = 1$ and $f, g \in L^1(\mu )$ are comonotone, show that

Let $f : \mathbb {R}^+ \to \mathbb {R}^+$, $\mathbb {R}^+ = \{ x \in \mathbb {R} : x > 0\}$. Assume that $f \in L^1(\mathbb {R}^+)$ satisfies

with $c$ independent of $x$. Show that $f(x) = 0$ for every $x \in \mathbb {R}^+$.

Show that the following limit exists

A Lebesgue integrable function $f : \mathbb {R} \to \mathbb {R}$ has the property that

for all Lebesgue measurable sets $E \subset \mathbb {R}$ with $m(E) = \pi$. Prove or disprove that $f = 0$ a.e.

Prove that the following limit exists

and find it, justifying all your steps.

Let $(X, \mathcal{X}, \mu )$ be a measure space and let $f_k : X \to \mathbb {R}$ be a sequence of measurable functions on it satisfying

where $c$ is a finite constant independent of $k$, and

For each $n = 1, 2, \ldots$, set $g_n = \sum_{k=1}^{n} f_k$. Prove that

for all $\alpha > 3/2$.

Let $(S, \Omega , m)$ be a measure space satisfying $m(S) < \infty$. Let $f : S \to \mathbb {R}$ be a measurable function satisfying $\left|f\right| \leq 1$. Prove that either $\lim_{n \to \infty } \int_S (1 + f + f^2 + \cdots + f^n) \, dm$ exists or that this limit is $+\infty$.

If $\{ r_k\}$ denotes the rational numbers in $[0,1]$ and $\{ a_k\}$ satisfies $\sum_{k=1}^{\infty } \left|a_k\right| < \infty$, prove that $\sum_{k=1}^{\infty } \dfrac {a_k}{\left|x-r_k\right|^{1/2}}$ converges absolutely a.e.$[0,1]$.

Prove that the series

is summable and find its sum.

Let $\{ a_{n,k} \mid n, k = 1, 2, \ldots \} \subset \mathbb {R}$ with $\left|a_{n,k}\right| \leq 1$ for $n, k \geq 1$. Assume that for each $n$, $\lim_{k \to \infty } a_{n,k} = 0$. Let $p > 1$. Show that $\lim_{k \to \infty } \sum_{n=1}^{\infty } \dfrac {a_{n,k}}{n^p} = 0$.

Prove that if $f$ and $g$ are positive continuous functions on $(-\infty , +\infty )$ which are periodic of period $1$ that

Given functions $\{ f_n : [0,1] \to [-1,1]\}$ such that $\lim_{n \to \infty } \int_a^b f_n(x) \, dx = 0$ for all $0 \leq a < b \leq 1$, show that

Let $f$ be a Lebesgue integrable function on $\mathbb {R}^2$, that is $\int_{\mathbb {R}^2} \left|f(x)\right| \, dx < \infty$. Assume that $\int_B f(x) \, dx = 0$ for all square subsets $B \subset \mathbb {R}^2$. Prove that $f(x) = 0$ almost everywhere.

Let $f \in L^1[0,1]$ and $\varepsilon > 0$. Prove the existence of a compact set $K$ and a polynomial $p$ such that the Lebesgue measure of $K$ exceeds $1 - \varepsilon$ and

Assume that $f_n \leq g_n \leq h_n$ are real-valued Lebesgue integrable functions converging almost everywhere to Lebesgue integrable functions $f, g$ and $h$ respectively. Assume that $\lim_{n \to \infty } \int f_n(x) \, dx = \int f(x) \, dx$ and $\lim_{n \to \infty } \int h_n(x) \, dx = \int h(x) \, dx$. Prove that $\lim_{n \to \infty } \int g_n(x) \, dx = \int g(x) \, dx$.

Let $f_n : [0,1] \to [0,\infty )$ be Lebesgue integrable for $n = 1, 2, \ldots$. Assume that

Let $g(x) = \sup \{ f_n(x) : n \geq 1\}$. Prove that $\int_0^1 g(x) \, dx = \infty$.

Let $\lambda$ be Lebesgue measure and $\mu$ be counting measure both regarded as Borel measures on $I = [0,1]$. Let $\Delta = \{ (x,x) \mid x \in I\}$, the diagonal of $I \times I$.