Ruby Briggs

2022-07-20

Prove that ${\chi}_{(0,1)}-{\chi}_{S}$ is not a limit of increasing step functions.

Let ${A}_{n}$ be sequence of connected bounded subsets (interval) of real numbers. Step function is defined to be the finite linear combination of their charateristic functions.

$\psi ={c}_{1}{\chi}_{{A}_{1}}+{c}_{2}{\chi}_{{A}_{2}}+...+{c}_{n}{\chi}_{{A}_{n}}$

while ${c}_{k}\in \mathbb{R}$ for $k=1,2,...,n$.

Let ${I}_{n}$ be a sequence of open intervals in (0,1) which covers all the rational points in (0,1) and such that $\sum \int {\chi}_{{I}_{n}}\le \frac{1}{2}$. Let $S=\bigcup {I}_{n}$ and $f={\chi}_{(0,1)}-{\chi}_{S}$ show that there is no increasng sequence of step function $\{{\psi}_{n}\}$ such that $lim{\psi}_{n}(x)=f(x)$ almost everywhere. (by means of increasing, ${\psi}_{n}(x)\le {\psi}_{n+1}(x)$ for all x)

I think ${\psi}_{n}={\chi}_{(0,1)}-\sum _{k=1}^{n}{\chi}_{{I}_{k}}$ is what author intended. $\int {\psi}_{n}$ is decreasing and $\int {\psi}_{n}\ge 1-\frac{1}{2}$. this shows that $lim\int {\psi}_{n}$ converges. so $lim{\psi}_{n}(x)=f(x)$ almost everywhere. However convergence of ${\psi}_{n}$ doesn't prove the non-existence of increasing step function which converges to f a.e.

How can I finish the proof?

Let ${A}_{n}$ be sequence of connected bounded subsets (interval) of real numbers. Step function is defined to be the finite linear combination of their charateristic functions.

$\psi ={c}_{1}{\chi}_{{A}_{1}}+{c}_{2}{\chi}_{{A}_{2}}+...+{c}_{n}{\chi}_{{A}_{n}}$

while ${c}_{k}\in \mathbb{R}$ for $k=1,2,...,n$.

Let ${I}_{n}$ be a sequence of open intervals in (0,1) which covers all the rational points in (0,1) and such that $\sum \int {\chi}_{{I}_{n}}\le \frac{1}{2}$. Let $S=\bigcup {I}_{n}$ and $f={\chi}_{(0,1)}-{\chi}_{S}$ show that there is no increasng sequence of step function $\{{\psi}_{n}\}$ such that $lim{\psi}_{n}(x)=f(x)$ almost everywhere. (by means of increasing, ${\psi}_{n}(x)\le {\psi}_{n+1}(x)$ for all x)

I think ${\psi}_{n}={\chi}_{(0,1)}-\sum _{k=1}^{n}{\chi}_{{I}_{k}}$ is what author intended. $\int {\psi}_{n}$ is decreasing and $\int {\psi}_{n}\ge 1-\frac{1}{2}$. this shows that $lim\int {\psi}_{n}$ converges. so $lim{\psi}_{n}(x)=f(x)$ almost everywhere. However convergence of ${\psi}_{n}$ doesn't prove the non-existence of increasing step function which converges to f a.e.

How can I finish the proof?

Hassan Watkins

Beginner2022-07-21Added 18 answers

Explanation:

Suppose such a sequence $\{{\psi}_{n}\}$ exists. Note that ${\psi}_{n}\le 0$ at rational points since $f\le 0$ at those points. If a step function is non-positive at rational points it is so at all but countable many points. [ The countable many points I am referring to are the end points of teh intervals on which the function is a constant]. It follows that $\int {\psi}_{n}\le 0$ and hence $\int f\le 0$ which is clearly a contradiction.

Suppose such a sequence $\{{\psi}_{n}\}$ exists. Note that ${\psi}_{n}\le 0$ at rational points since $f\le 0$ at those points. If a step function is non-positive at rational points it is so at all but countable many points. [ The countable many points I am referring to are the end points of teh intervals on which the function is a constant]. It follows that $\int {\psi}_{n}\le 0$ and hence $\int f\le 0$ which is clearly a contradiction.

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4), and D(5, -1)?

How to expand and simplify $2(3x+4)-3(4x-5)$?

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

How to graph $r=5\mathrm{sin}\theta$?

How to find the length of a curve in calculus?

When two straight lines are parallel their slopes are equal.

A)True;

B)FalseIntegration of 1/sinx-sin2x dx

Converting percentage into a decimal. $8.5\%$

Arrange the following in the correct order of increasing density.

Air

Oil

Water

BrickWhat is the exact length of the spiraling polar curve $r=5{e}^{2\theta}$ from 0 to $2\pi$?

What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?

What is the slope of the tangent line of $r=-2\mathrm{sin}\left(3\theta \right)-12\mathrm{cos}\left(\frac{\theta}{2}\right)$ at $\theta =\frac{-\pi}{3}$?

How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?

Use the summation formulas to rewrite the expression $\Sigma \frac{2i+1}{{n}^{2}}$ as i=1 to n without the summation notation and then use the result to find the sum for n=10, 100, 1000, and 10000.

How to calculate the right hand and left hand riemann sum using 4 sub intervals of f(x)= 3x on the interval [1,5]?