--- title: "Conditional Expectation Given a Sigma-Algebra" description: "Define conditioning on information rather than on an event, as the best prediction measurable with respect to what is known." canonical: https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8 source: https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Conditional Expectation Given a Sigma-Algebra Define conditioning on information rather than on an event, as the best prediction measurable with respect to what is known. Subject: Mathematics · Area: Probability · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8 ## Ready when they can - State the defining property and prove uniqueness up to a null set - Recover elementary conditional expectation as the case of a finite partition - Use the tower property and the projection interpretation on a worked example ## Lesson: Forecasting with what you know Conditioning on a sigma algebra means conditioning on information rather than on a single event. The conditional expectation E of X given G is a G measurable forecast: it uses only what G knows. Its defining property is that it matches X in the mean on every G set, through E of indicator A times E of X given G equals E of indicator A times X. Two versions can differ only on null sets, which integrals cannot tell apart, so uniqueness holds up to null sets. **Example.** Roll a fair six sided die and let G record only whether the roll is even or odd. On the even block the forecast is the local mean 2 plus 4 plus 6 over 3, which is 4. On the odd block it is 1 plus 3 plus 5 over 3, which is 3. That is the finite partition case in general: condition on each block and report the average within it. The tower property says averaging the forecast recovers the plain mean: E of E of X given G equals E of X. With two fair coin flips where G records only the first flip, the forecast averages back to 1, the mean number of heads. When X is independent of G, conditioning changes nothing and E of X given G equals E of X. The projection view says the same thing geometrically: E of X given G is the closest G measurable guess in mean square. **Tip.** When you verify a candidate forecast, check the two halves separately: measurability with respect to G, then the defining identity on every G set. Students usually check one half and forget the other. If either half fails, the candidate is not the conditional expectation. **Recap.** Condition on what you know, match the mean on every known set, and average the forecast to recover the mean. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Sigma-Algebras and Measurable Sets](https://lightmysky.com/learn/mathematics/sigma-algebras-and-measurable-sets-mt_cpvegazufk) - [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU) - [Hilbert Spaces and Orthogonal Projection in Infinite Dimensions](https://lightmysky.com/learn/mathematics/hilbert-spaces-and-orthogonal-projection-in-infinite-dimensions-mt_xmKq8SAgvy) ## Opens up - [Martingales and Optional Stopping](https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ) - [Markov Chains and the Memoryless Assumption](https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o)