Decompose $[0,1]$ into countably many non-measurable sets . Can any one give me any idea about how to decompose $[0,1]$ into countably many non-measurable sets? Thanks.

Here is another method:

Find out a way to divide an interval into two non-measurable sets (which means that you should know how to find _one_ non-measurable set, and then its complement is also non-measurable).

Next divide $[0,1]$ into the following parts: $\\{0\\}$ and the rest we split into intervals $\left(\frac1{n+1},\frac1n\right]$. Each of these intervals we cut to two non-measurable parts, and we know how to do that. Lastly, the singleton $\\{0\\}$ should be added to one of these non-measurable sets, and we are done.