LetY be a random variable and define the error term by u = Y E(Y|(X)). Show that by definition, this random variable satisfies the following

LetY be a random variable and define the error term by u = Y −E(Y|σ(X)). Show that by definition, this random variable satisfies the following properties:

[i] E(u|σ(X)) = 0, [ii] E(u·X|σ(X)) = 0,

[iii] E(u) = 0, [iv] E{u·[E(Y|σ(X)]|σ(X)} = 0.