Appproximating Functions by Step Functions

Question

Q.1 For the Dirichlet function (=characteristic function of rationals in $[0,1]$), does there exists an increasing sequence of step functions which converges point-wise to the Dirichlet function?

Q.2 Is it true that a (non-negative) function $f\colon [a,b]\rightarrow \mathbb{R}$ is Riemann integrable if an only if there exists an increasing sequence of step functions converging point-wise to $f$?

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Step function on $[a,b]$ means piecewise constant function, i.e. $[a,b]$ can be partitioned into finitely many sub-intervals (closed) such that the function is constant on each sub-interval, except at end-points of sub-interval.

Answer 1

If your definition of interval is nondegenerate interval (that is, $a<b$) then the answer is no.

If you allow degenerate intervals (a point), the answer is yes. Let $$ A_n=\left \{\frac pq:p,q\in\mathbb{Z},\ 1\le q\le n, \ 0\le p\le q \right \} $$ and let $\chi_n$ be its characteristic function. Then $\chi_n$ is increasing (because $A_n\subset A_{n+1}$) and converges pointwise to Dirichlet's function.

Answer 2

Suppose there exists an increasing sequence of step functions $\{s_n\}$ which converges point-wise to the Dirichlet function $f$.

Let $q\in \mathbb{Q}$, then $\exists N\in \mathbb{N}$, s.t. $s_N(q)>\frac{1}{2}$. Since it's a step function, $\exists x\in [0,1]\setminus \mathbb{Q}$ closed enough to $q$, s.t $s_N(x)=s_N(q)>\frac{1}{2}$. Since it's an increasing sequence, $s_n(x)>\frac{1}{2} \forall n\geq N$. But pointwise convergence, $f(x)\geq \frac{1}{2}$. We get a contradiction since Dirichlet function has $0$ on irrational number.

Here is another example if you allow degenerate interval. Since $\mathbb{Q}$ is countable set, we order it as a sequence $\{q_1,q_2,q_3 \cdots\}$. Then we set $\varphi_n(x)=1$ for $x=q_1,q_2,\cdots q_n$ and $0$ otherwise. Then we get an increasing sequence and pointwise converges to Dirichlet function.

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For Q2, I think it is not correct. We can consider the Thomae's function, which is integrable and apply a similar argument as Q1.