Section II.2

Measurable Functions

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Subsection II.2.1Properties of Measurable Functions[18/78%]

Problem 125

Let $X$ be a measure space and $f$ a real-valued function defined on $X$ . Prove that the following properties of $f$ are equivalent:

for any $a\in \mathbb {R}$ the set $\left\{ x\in X:f(x)>a\right\}$ is measurable;
for any $a\in \mathbb {R}$ the set $\left\{ x\in X:f(x)\geq a\right\}$ is measurable;
for any $a\in \mathbb {R}$ the set $\left\{ x\in X:f(x)<a\right\}$ is measurable;
for any $a\in \mathbb {R}$ the set $\left\{ x\in X:f(x)\leq a\right\}$ is measurable.
$f^{-1}(B)$ is a measurable set for any Borel set $B\subset \mathbb {R}$ .

Problem 126

Suppose that $f$ is measurable and does not vanish. Prove that $1/f$ is a measurable function.

Problem 127

Prove that $\left|f\right|$ is a measurable function if $f$ is.

Problem 128

Let $f(t_1,t_2,\ldots ,t_n)$ be a continuous real-valued function defined on a real $n$ -dimensional space, and $g_1(x),\ldots ,g_n(x)$ measurable functions. Prove that $h(x)=f(g_1(x),\ldots ,g_n(x))$ is a measurable function.

Problem 129

Suppose that $g(x)$ is a measurable function defined on the real line, and $f$ is a continuous real function. Show that $h(x)=g(f(x))$ need not be a measurable function.

Problem 130

Let $f(x)$ be a function differentiable everywhere on $[0,1]$ . Prove that $f'(x)$ is Lebesgue measurable.

Problem 131

Let $f(x)$ be the Cantor one-to-one mapping of $[0,1]$ onto the square: $f(x)=(y_1,y_2)$ with $y_1=0.x_1x_3\ldots$ and $y_2=0.x_2x_4\ldots$ for $x=0.x_1x_2x_3\ldots$ . Prove that $f(x)$ carries any measurable subset of the interval into a measurable subset of the square and preserves the value of measure.

Problem 132

A function $f(x)$ defined on the real line is called a Borel function if $\left\{ x\in \mathbb {R}:f(x)<a\right\}$ is a Borel set for any $a\in \mathbb {R}$ (see Problem 99(b)). Prove that any measurable function becomes a Borel function after suitable modification on a set of measure zero.

Problem 133

Let $\mu$ be Lebesgue measure on $[0,1]$ , and $f$ a measurable function that is finite almost everywhere on this segment. Prove that for any $\varepsilon >0$ there is a closed set $F\subset [0,1]$ such that the restriction of $f$ to $F$ is continuous and $\mu \left(F\right)>1-\varepsilon$ .

Problem 134

Let $f$ be a measurable function defined on the real line. A point $x\in \mathbb {R}$ is called a Lebesgue point of $f$ if there is a Lebesgue measurable subset $X\subset \mathbb {R}$ that contains $x$ , has $x$ as a point of density, and is such that the restriction $f|_X$ is continuous at $x$ . Prove that almost every point of the line is a Lebesgue point for $f$ .

Problem 135

Let $f(x)$ be a continuous function defined on $[a,b]$ , and $n(c)$ the number of solutions of the equation $f(x)=c$ . Prove that $n(c)$ is Lebesgue measurable.

Problem 136

Let $f_n(x)$ be a sequence of measurable functions. Prove that the functions $\sup_n f_n(x)$ and $\inf_n f_n(x)$ are measurable.

Problem 137

Let $\left\{ f_n\right\}$ be a sequence of measurable functions. Prove that the set of all points $x$ where $\lim_{n\to \infty }f_n(x)$ exists is measurable.

Problem 138

Let $f$ be a measurable function. Prove that its positive part $f^+=\max (f,0)$ and its negative part $f^-=-\min (f,0)$ are measurable functions.

Problem 139

(a)(b)

Real functions $f$ and $g$ that are measurable with respect to the respective measures $\mu$ and $\nu$ are said to be equimeasurable if $\mu \left(x:f(x)<c\right)=\nu \left(y:g(y)<c\right)$ for any $c>0$ .

(a)

Prove that if $f$ is measurable with respect to a measure $\mu$ , then there exists a left-continuous nondecreasing function $g$ on the segment $[0,\mu \left(X\right)]$ that is equimeasurable with $f$ .

(b)

Under the conditions of Problem 139(a) prove that $g(x)$ is unique.

Problem 140

A complex-valued function $f(x)=u(x)+iv(x)$ is said to be measurable if its real part $u(x)$ and imaginary part $v(x)$ are measurable. Prove that the modulus and argument of $f(x)$ are measurable.

Problem 141

Prove that a complex-valued function $f(x)$ is measurable if and only if all the sets of the form $A_{r,z}=\left\{ x:\left|f(x)-z\right|\leq r\right\}$ are measurable, where $z\in \mathbb {C}$ , $r\geq 0$ .

Problem 142

A vector-valued function $f$ with values in a finite-dimensional space $V$ is said to be measurable if the coordinates of $f(x)$ with respect to some basis in $V$ are measurable. Prove that this definition does not depend on the choice of the basis.

Subsection II.2.2Convergence of Measurable Functions[19/84%]

Problem 143

Prove that the sequence $f_n(x)=nx/(n^2+x^2)$ converges to zero everywhere on $\mathbb {R}$ , but not uniformly.

Problem 144

Investigate the convergence and uniform convergence of the sequence $f_n(x)=x^n$ on $[0,1]$ .

Problem 145

Prove that two continuous functions on a segment are equivalent with respect to Lebesgue measure only if they are identically equal.

Problem 146

Construct on a closed interval a Lebesgue-measurable function that is not equivalent to any continuous function.

Problem 147

Show that if $f_n\xrightarrow {\mathrm{a.e.}}f$ and $f_n\xrightarrow {\mathrm{a.e.}}g$ , then $f$ is equivalent to $g$ .

Problem 148

(a)(b)

Let the set of rational numbers in $[0,1]$ be enumerated, with the $k$ th number $r_k$ in the form of an irreducible fraction $r_k=p_k/q_k$ . Define $f_k(x)=\mathrm{exp}\left(-(p_k-xq_k)^2\right)$ .

(a)

Prove that $f_k\to 0$ in Lebesgue measure on $[0,1]$ , and that $\lim_{n\to \infty }f_k(x)$ does not exist at any point of the segment.

(b)

Under the conditions of Problem 148(a) determine explicitly a subsequence converging to zero almost everywhere.

Problem 149

Define functions $f_i^{(k)}$ on $[0,1]$ by setting

f_i^{(k)}(x)= \begin{cases} 1 & \text{for }(i-1)/k\leq x<i/k,\quad i=1,2,\ldots ,k,\ k=1,2,\ldots ,\\ 0 & \text{otherwise}, \end{cases}

and let $g_n(x)=f_i^{(k)}(x)$ , where $i$ and $k$ are determined by the condition $n=k(k-1)/2+i$ . Prove that $g_n\to 0$ in measure, but $\lim g_n(x)$ does not exist at any point.

Problem 150

Suppose that $f_n\xrightarrow {\mu }h$ and $f_n\xrightarrow {\mu }g$ . Prove that $h$ and $g$ are equivalent with respect to the measure $\mu$ .

Problem 151

Prove that a real function on $[a,b]$ is Lebesgue-measurable if and only if for any $\varepsilon >0$ there is a continuous function that differs from $f$ on a set of measure $\leq \varepsilon$ .

Problem 152

It follows from Luzin's theorem (Problem 151) that every measurable function $f$ on $[a,b]$ is a limit almost everywhere of a sequence $\left\{ f_n\right\}$ of continuous functions. Is it always possible to choose this sequence to be monotone?

Remark.

By using Levi's theorem, it is easy to prove that no measurable function $f(x)=f_+(x)-f_-(x)$ such that $\int_{[a,b]}f_+(x)\, dx=\int_{[a,b]}f_-(x)\, dx=+\infty$ can be represented as a monotone limit of a sequence of integrable functions.

Theorem 16 (B. Levi's theorem on monotone convergence, §3, Chapter II): Let $\left\{ f_n\right\}$ be a monotonically increasing sequence of $\mu$ -integrable functions on $X$ , and let $f(x)=\lim_{n\to \infty }f_n(x)$ (the value $+\infty$ is allowed).

If the integrals $\int_X f_n(x)\, d\mu (x)$ are collectively bounded, then $f$ is integrable and $\int_X f(x)\, d\mu (x)=\lim_{n\to \infty }\int_X f_n(x)\, d\mu (x)$ .
If $\lim_{n\to \infty }\int_X f_n(x)\, d\mu (x)=+\infty$ , then $f$ is not integrable.

Problem 153

(a)(b)

(a)

(b)

The Dirichlet function

\psi (x)= \begin{cases} 0 & \text{if }x\text{ is irrational},\\ 1 & \text{if }x\text{ is rational} \end{cases}

can be obtained from continuous functions by the iterated limit

\psi (x)=\lim _{n\to \infty }\lim _{m\to \infty }[\cos (2\pi n!x)]^m.

Is it possible to obtain it from continuous functions by a single limit?

Remark.

(b)

Noncontinuous functions representable as the limit of a sequence of continuous functions are called functions of the 1st Baire class. The Dirichlet function belongs to the 2nd Baire class (double limit).

Problem 154

Prove that a simple function (i.e., a function taking not more than countably many values) is measurable if and only if all its level sets

L_c(f)=\left\{ x\in X:f(x)=c\right\}

are measurable. Is this true for arbitrary functions?

Problem 155

Prove that every measurable function can be represented as a uniform limit of measurable simple functions.

Problem 156

(a)(b)

Define the function $f(x)$ on $[0,1]$ as follows. If $x=0.n_1n_2n_3\ldots$ is the decimal expansion of $x$ , then $f(x)=\max_i n_i$ .

(a)

Prove that $f(x)$ is measurable and almost everywhere constant.

(b)

Under the conditions of Problem 156(a) show that the function $f(x)=\limsup_{i\to \infty }n_i$ is defined everywhere and constant almost everywhere.

Problem 157

Let $\mu$ be Wiener measure on $X=C[a,b]$ . Define a function $f$ on $X$ by setting $f(x)=\int_a^b x(t)\, dt$ . Prove that $f$ is $\mu$ -measurable.

Remark.

The Wiener measure in §1.3 of Chapter II is the Lebesgue extension of the measure on the half-ring of cylinder sets

X(t_1,\ldots ,t_n;\Delta _1,\ldots ,\Delta _n)=\left\{ x\in C[a,b]:x(t_k)\in \Delta _k\text{ for }k=1,\ldots ,n\right\} ,

where $t_1\leq \cdots \leq t_n$ are points in $[a,b]$ and $\Delta_1,\ldots ,\Delta_n$ are intervals on the real line. Its values are

\begin{align} \mu \left(X(t_1,\ldots ,t_n;\Delta _1,\ldots ,\Delta _n)\right)={}& \pi ^{(1-n)/2}\prod _{k=1}^{n-1}(t_{k+1}-t_k)^{-1/2}\\ & \times \int _{\Delta _1}\cdots \int _{\Delta _n}\mathrm{exp}\left(-\sum _{k=1}^{n-1}\frac{(\tau _{k+1}-\tau _k)^2}{t_{k+1}-t_k}\right)\, d\tau _1\cdots d\tau _n. \end{align}

For $n=1$ , this means $\mu \left(X(t,\Delta )\right)=\int_\Delta d\tau =\left|\Delta \right|$ .

Problem 158

(a)(b)

Let $\mu$ be Wiener measure on $X=C[a,b]$ . Prove that $f$ is $\mu$ -measurable.

(a)

Do the same for the function $f(x)=\int_a^b\varphi (x(t),t)\, dt$ , where $\varphi (x,y)$ is a continuous function of two variables.

(b)

Do the same for the function $f(x)=\max_{t\in [a,b]}x(t)$ .

Problem 159

Let $X$ be the set of $p$ -adic integers (see Problem 33), $S$ the algebra of subsets of $X$ that are simultaneously open and closed in $X$ , and $\mathfrak A=R_\sigma (S)$ . Prove that every continuous function on $X$ is $\mathfrak A$ -measurable.

Problem 160

Under the conditions of Problem 159 prove that every set $A\in S$ is a union of finitely many balls. Define a measure $\mu$ on $S$ by setting the measure of a ball equal to its radius (for balls of radius $p^{-k}$ , $k=0,1,2,\ldots$ ). Prove that the measure $\mu$ is countably additive.

Problem 161

Prove that the measure $\mu$ in Problem 160 has the properties:

$\mu \left(X\right)=1$ ;
$\mu \left(A+x\right)=\mu \left(A\right)$ for all $x\in X$ . Prove that every measure on $\mathfrak A$ having the properties (a) and (b) coincides with $\mu$ .