Examples of non-Riemann integrable functions besides unbounded and Dirichlet functions.

Question

Everywhere I've tried to look, the only two common examples of non-Riemann integrable functions are unbounded functions or Dirichlet function.

What are some examples of non-Riemann integrable functions besides those functions?

I assume for such non-Riemann integrable functions, the lower integral and the upper integral must exist but are unequal. (Because the lower integral and upper integral both either exist or don't, together and since I have removed unbounded functions from the picture, both integrals exist but are unequal.)

I'd really appreciate multiple examples, if that's possible, of functions that have unequal lower and upper integral.

Answer 1

You can take, for instance$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x&\text{ if }x\in\Bbb Q\\0&\text{ if }x\notin\Bbb Q.\end{cases}\end{array}$$It is bounded. And it is not Riemann integrable since it is discontinuous at every point of $(0,1]$, and this is is uncountable. Another possibility would be$$\begin{array}{rccc}g\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x&\text{ if }x\in\Bbb Q\\-x&\text{ if }x\notin\Bbb Q.\end{cases}\end{array}$$

Answer 2

Let $A\subset [0, 1]$ be any set.

$\mathbf{1}_A:[0,1]\to\Bbb{R}$ defined by

$\mathbf{1}_A(x)=\begin{cases}1&x\in A\\ 0& x\notin A\end{cases}$

$\mathbf{1}_A$ is called indicator function of $A$.

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Set of discontinuity of $\mathbf{1}_A=\partial(A)$

where $\partial(A)$ denote the set of all boundary points of $A$

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Lebesgue's criteria for Riemann integrability : $f:[a,b]\to \Bbb{R}$ is Riemann integrable if and only if $f$ is bounded, and the set of discontinuities has Lebesgue measure zero.

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Step $1$ : Choose $A\subset [0, 1]$ such that $m(\partial (A)) >0$

Where $m$ is the $1$-dimensional Lebesgue measure.

Step $2$: Then choose $\mathbf{1}_A$

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Few examples: (Bounded functions which are not Riemann integrable)

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1. Choose the set of rationals $Q$ . Then $m(\partial({\Bbb{Q}\cap [0,1]}))=[0, 1]=1>0$

Hence $\mathbf{1}_{\Bbb{Q}} : [0, 1]\to \Bbb{R}$ defined by $\mathbf{1}_A(x)=\begin{cases}1&x\in \Bbb{Q}\\ 0& x\notin \Bbb{Q}\end{cases}$

(It is your known example, Dirichlet function)

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2. Choose the set of irrationals $A=\Bbb{R}\setminus \Bbb{Q}$.

Then $\mathbf{1}_{{A}} : [0, 1]\to \Bbb{R}$ defined by $\mathbf{1}_{A}(x)=\begin{cases}1&x\in \Bbb{Q}\\ 0& x\notin \Bbb{Q}\end{cases}$

Another bounded function which is not Riemann integrable.

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3. Choose the indicator function of $S\subset [0, 1] $ Cantor set of positive measure ( Specifically S-V-C set of measure $\frac{1}{2}$)

4. Choose the indicator function of $A=\{\sin n :n\in \Bbb{N}\}$

5. Choose the indicator function of $A=\Bbb{Q}\cup \{\text{ few of your favorite irrationals!}\}$

There are many more. Can you list few more?