The goal is to prove that $\mathbb{E}[L^2 f(x)] = \mathbb{E}[L f(x)]$, where $L$ is a linear operator and $\mathbb{E}$ represents the expectation.

Assumptions

1. The linear operator $L$ acts on the function $f(x)$, where $f(x)$ is some function.
2. The expectation operator $\mathbb{E}$ is linear.

Proof Outline

1. Linearity of the Operator $L$:
   A linear operator $L$ satisfies: $L(a f(x) + b g(x)) = a L(f(x)) + b L(g(x))$ for any functions $f(x), g(x)$ and scalars $a, b$.

2. Applying the Operator Twice: Since we are dealing with $L^2$, we are applying the operator $L$ twice: $L^2 f(x) = L(L(f(x))).$

3. Expectation of $L^2 f(x)$:
   Now consider the expectation: [ \mathbb{E}[L^2 f(x)] = \mathbb{E}[L(L(f(x)))]. ]

4. Commuting the Expectation and Linear Operator:
   If $L$ and $\mathbb{E}$ commute, i.e., if the linear operator $L$ can be taken outside the expectation (this is typically true for many linear operators under regularity conditions), then we can write: [ \mathbb{E}[L(L(f(x)))] = L(\mathbb{E}[L(f(x))]). ]

5. Final Step:
   Since $\mathbb{E}[L(f(x))] = L(\mathbb{E}[f(x)])$, we can write: [ \mathbb{E}[L^2 f(x)] = L(\mathbb{E}[L(f(x))]) = \mathbb{E}[L f(x)]. ]

Thus, we have shown that: [ \mathbb{E}[L^2 f(x)] = \mathbb{E}[L f(x)]. ]

Assumptions:

• This proof assumes that $L$ and $\mathbb{E}$ commute and that $L$ is a linear operator.

Would you like more details or have any specific questions?

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Here are 5 questions to expand on this concept:

1. How does linearity of the expectation operator play a role in this proof?
2. Under what conditions do the linear operator $L$ and the expectation $\mathbb{E}$ commute?
3. Can you provide an example where $L$ is a differential operator? How would this affect the proof?
4. What changes in the proof if $L$ is not linear?
5. How would this result differ if the expectation operator were taken after applying $L^2 f(x)$?

Tip: When working with linear operators, understanding how they interact with integrals or sums (like expectations) is crucial to simplifying complex expressions.