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  "body": "\n## Proof\n\nWe apply the defining property of the conditional expectation to the whole space.\n\n1. Let $Z = E[X \\mid \\mathcal{G}]$ be a [[conditional expectation|0YAMJRK04E]] of $X$ given $\\mathcal{G}$. By definition, $Z$ is integrable and $\\mathcal{G}$-measurable.\n2. The definition also gives $E[Z \\, 1_G] = E[X \\, 1_G]$ for each $G \\in \\mathcal{G}$.\n3. A [[σ-algebra|04WSW8T3SY]] on $\\Omega$ contains $\\Omega$. So $\\Omega \\in \\mathcal{G}$, and step 2 applies with $G = \\Omega$.\n4. The indicator $1_\\Omega$ is the constant $1$. So step 3 reads $E[Z] = E[X]$, where $E$ is the [[expected value|0DCT0QZVQ1]].\n5. Two versions of $E[X \\mid \\mathcal{G}]$ are equal almost surely, so they have the same expectation. Thus step 4 does not depend on the choice of $Z$, and $E[E[X \\mid \\mathcal{G}]] = E[X]$.\n6. Let $Y$ be a random variable on the same space. By definition, $E[X \\mid Y] = E[X \\mid \\sigma(Y)]$, and $\\sigma(Y) \\subseteq \\mathcal{F}$ is a σ-algebra.\n7. Step 5 with $\\mathcal{G} = \\sigma(Y)$ gives $E[E[X \\mid Y]] = E[X]$.\n\n## Idea\n\nThe conditional expectation is built so that it has the same average as $X$ on every event of $\\mathcal{G}$. The whole space $\\Omega$ is one of these events, and the average over $\\Omega$ is the ordinary expectation.\n",
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