Why do we essentially need complete measure space?

Solution 1:

We want measure spaces to be complete because we want to treat sets of measure zero as negligible. For example, if two functions $f$ and $g$ satisfy $f(x)=g(x)$ for all $x\in X\setminus N$, and $N$ has measure zero, then we'd like to treat $f$ and $g$ as essentially the same thing. However, without completeness it's possible that $f$ is measurable but $g$ is not.

The issue of completeness is brought into light by the product operation, because the product of complete measures is not always complete. For example, let $A\in [0,1]$ be a nonmeasurable set. The set $A\times \{0\}\subset [0,1]\times [0,1]$ is not measurable with respect to the product measure $\lambda\otimes\lambda$. However, $A\times \{0\}\subset [0,1]\times \{0\}$ and the latter set has product measure $0$. So, once we take the completion of the product measure, $A\times \{0\}$ becomes a measure $0$ set.

Solution 2:

I am giving an answer of this question Philosophically...

All intuition of Measure Theory comes from Probability Theory (finite measure theory ). In Probability certain event is impossible then all its sub events are also impossible (usually).