Step function and simple function

Question

I am trying to understand clearly whether there is any relation between the set of step function and simple function. Till now I have got that none is the subset of other. The Dirichlet function is a simple function(correct me if i am wrong.) Which is not a step function. I am not able to conclude clearly. Kindly help me with this.

Answer 1

A simple function by definition is a function taking only finitely many values. A step function is a finite linear combination of indicator functions of intervals.
In particular, each step function takes finitely many values, hence is a simple function. The other implication does not hold.

Answer 2

Step functions are simple (because intervals are measurable), not all simple functions are step functions (because of your example for instance). More conceptually, step functions are also sort of the Riemann analogue of Lebesgue's simple functions, in that Riemann integrals are obtained by integrating a sequence of step functions converging pointwise to the true integrand.