Monotonically approximate $L^p$ function by step function

Question

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is monotonically increasing (suppose that $f>0$ a.e. or we can do it for $f^\pm$ respectively)?

Answer 1

I expect you mean to ask if you can find a monotonically increasing sequence of step functions approximating the given function from below?

The answer is a resounding no. For a simple example, consider the characteristic function of a fat Cantor set.

Answer 2

Looking at non-negative unsigned Lebesgue integrable functions on $\mathbb{R}$,

• step function: taking finitely many values on finitely many intervals.
• simple function: taking finitely many values on finitely many Lebesgue measurable sets.

True: Both step functions and simple functions are dense in $L^1$. They can approximate any Lebesgue integrable function $f$ arbitrarily close using the integral norm $||\cdot||_{L^1}$.

True: For each unsigned Lebesgue integrable function $f$, there exists a sequence of monotone increasing simple functions $g_n$ such that $g_n\rightarrow f$ in $L^1$.

As Harald Olsen said, even step functions are dense in $L^1$, the statement about monotone is false. And here is the intuitive idea:

The pre-image of a step function has to be intervals. But given a Lebesgue measurable set $A$, if only using intervals, some times we can only approximate $A$ from the outside but not from the inside. That is $$\mu(A) = \inf\left\{ \sum_{n=1}^\infty |I_n| : A\subset \cup_n I_n \right\}$$ but $$\mu(A) \neq \sup\left\{ \sum_{n=1}^\infty |I_n| : A\supset \cup_n I_n \right\}.$$