Please excuse me if this is a silly question. I graduated from university many years ago and am now trying to re-learn some mathematics. I don't know much advanced mathematics and I have difficulty to see the big pictures in many aspects of mathematics.
Anyway, I am now learning Lebesgue and Daniell integrals. In a thread about Daniell scheme on this site, someone has quoted Vladimir Bogachev, who said that
...it is perfectly clear that the way of presentation in which the integral precedes measure can be considered as no more than equivalent to the traditional one.
However, in his book Integral, Measure and Derivative: a Unified Approach, Georgi Shilov wrote (on p.2) that
...it should be pointed out that the Lebesgue and Daniell constructions of the integral are equivalent if finite-valued ("step") functions are chosen as elementary functions. However, there are cases where functions other than step functions should be chosen as elementary functions (e.g., in studying linear functionals on the space of continuous functions defined on a compact metric space), and then the Daniell method is effective while the Lebesgue method is not.
He seemed to suggest that sometimes we can define a Daniell integral but not a Lebesgue integral.
Setting aside the question about the merits of each approach, can someone please clarify whether Lebesgue integration and Daniell integration are equivalent or not? Is one approach applicable whenever the other is applicable?
