Triangle Inequality for Conditional Expectation

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

Let $X$ be an integrable random variable.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Let $\expect {\size X \mid \GG}$ be a version of the conditional expectation of $\size X$ given $\GG$.

Then we have:

$\size {\expect {X \mid \GG} } \le \expect {\size X \mid \GG}$ almost everywhere.

Proof

From Conditional Expectation is Monotone, we have:

$\expect {X^+ \mid \GG} \ge 0$ almost everywhere

and:

$\expect {X^- \mid \GG} \ge 0$ almost everywhere

where $X^+$ and $X^-$ are the positive and negative parts respectively.

Now, almost everywhere we have:

\(\ds \size {\expect {X \mid \GG} }\)

\(=\)

\(\ds \size {\expect {X^+ - X^- \mid \GG} }\)

\(\ds \)

\(=\)

\(\ds \size {\expect {X^+ \mid \GG} - \expect {X^- \mid \GG} }\)

Conditional Expectation is Linear

\(\ds \)

\(\le\)

\(\ds \size {\expect {X^+ \mid \GG} } + \size {\expect {X^- \mid \GG} }\)

Triangle Inequality for Real Numbers

\(\ds \)

\(=\)

\(\ds \expect {X^+ \mid \GG} + \expect {X^- \mid \GG}\)

\(\ds \)

\(=\)

\(\ds \expect {X^+ + X^- \mid \GG}\)

Conditional Expectation is Linear

\(\ds \)

\(=\)

\(\ds \expect {\size X \mid \GG}\)

Sum of Positive and Negative Parts

$\blacksquare$