OK.
The average value of a function is the integral of the function divided by it's length. (Calculus stuff)
So, the definite integral of Sin(t) from 0 to PI is equal to 2. (1/2 of a cycle)
https://www.wolframalpha.com/input/?i=inegral sin(x) from 0 to PI
The sin(t) function has a max of 1, corresponding to Vp. in v(t)=Vp*sin(t)
Our full sin(t) goes from 0 to 2*PI, but let's invert (from 180 to 360 deg), so the integral of abs(sin(t)) from 0 to 2* PI = 4 for the rectified sin function.
So, the average is 4/2*PI for the sin wave and therefor 2*Vm/PI for the sine wave having Vmaximum of Vm.
Sorry, for the round-about way of getting there. My Calculus squeaks.
The real average value of a sine wave is zero, but the average of a rectified sine wave is 2*Vm/PI
I could have got there easier by saying a rectified sine wave has a period of PI and the definate integral of from 0 to PI of 2. Therefore, the average is 2/PI when Vm =1. When Vm <>1, it's 2Vm/PI.
The capacitor filter basically charges the cap to the peak value and if the cap isn't big enough or the input current is lower than required, it starts to decay.
The wierdness with average, also determines the max secondary draw. It's less than the RMS current rating of the secondary.