The whiteboard contains mathematical notes and equations. Here is the transcription:

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### Header:
"Measure-preserving probability spaces for dynamical systems, or equivalently \((X, \mu, T)\)"

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### Section 1 (Definitions and Notation):
1. \((X, \mathcal{B}, \mu)\) = a measure space
2. \( T: X \to X \) is measurable
3. \( T \) preserves \(\mu\): \(\mu(T^{-1}A) = \mu(A) \quad \forall A \in \mathcal{B}\)

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### Section 2 (Examples):
- \((X, \mu) = ([0, 1], \lambda)\), \( T(x) = 2x \mod 1\)
- Circle rotation \((X, \mu) = (\mathbb{S}^1, \lambda)\), \( T(x) = x + \alpha \mod 1\)

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### Section 3 (Ergodicity and Mixing):
#### Ergodicity:
1. \( T \) is **ergodic** if:
   - \(\forall A \in \mathcal{B}\), \(T^{-1}A = A \implies \mu(A) \in \{0, 1\}\)

#### Mixing:
2. \( T \) is **mixing** if:
   - \(\lim_{n \to \infty} \mu(T^{-n}A \cap B) = \mu(A)\mu(B)\)

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### Section 4 (Applications of Mixing):
- **Pointwise Ergodic Theorem**:
   - \(f \in L^1(X, \mu)\), \(\frac{1}{N} \sum_{n=1}^N f(T^n x) \to \int_X f \, d\mu \quad \mu\)-almost everywhere.

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### Section 5 (Further Notes and Symbols):
1. \( B: L^2(X, \mu) \to L^2(X, \mu) \), where \( B(f) = \int_X f d\mu \).
2. \( T \) generates a group \((T^n)_{n \in \mathbb{Z}}\).
3. Ergodic systems: No nontrivial invariant subsets.
4. Mixing implies ergodicity, but not vice versa.

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