Z-direction. Quantisation axis

[LagrangeEuler]

I'm a bit confused with this concept of quantisation axis. $|+\rangle$ and $|-\rangle$ spin. And operators $\hat{S}^+=\hat{S}_x+i\hat{S}_y$ and $\hat{S}^-=\hat{S}_x-i\hat{S}_y$ are rising and lowering operator of spin projection. Why z-axis. How is measured that?

 

[mfb]

Why z-axis.

The coordinate system is arbitrary, you could use any other axis as well. The coordinate system is just a mathematical tool, not something physical.

How is measured that?

What?