Conditional Expectation Mean square error and Orthogonality, 𝐸[𝑋𝑌|F]=𝑋𝐸[𝑌|F] | Martingale Theory

Conditional expectation properties: Linearity, Monotonicty, Chebyshev’s inequality, Markov inequality, Monotone convergence, Jenson inequality, Cauchy-Schwarz inequality, 𝐸[𝐸[𝑋|F]]=𝐸[𝑋].

Given F1⊂F2 then: 𝐸[𝐸[𝑋|F1]|F2]=𝐸[𝑋|F1].
Given F1⊂F2 then: 𝐸[𝐸[𝑋|F2]|F1]=𝐸[𝑋|F1]

Prove:
1. 𝐸[𝑋𝑌|F]=𝑋𝐸[𝑌|F]
2. Mean square error and Orthogonality