Monotonically approximate $L^p$ function by step function 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)?
 A: 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.
A: 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 
$$\mu(A) \neq \sup\left\{ \sum_{n=1}^\infty |I_n| : A\supset \cup_n I_n \right\}.$$
