To define a unique, satisfying expected value from chosen sequences of bounded functions converging to an everywhere surjective function, we must carefully interpret the convergence, surjectivity, and the expectation framework. Here’s a step-by-step conceptual breakdown:

1. Function Setting

Let {fn}\{f_n\} be a sequence of bounded functions fn:[0,1]→Rf_n: [0,1] \to \mathbb{R}, converging pointwise (or in some sense) to a function ff, where ff is everywhere surjective (i.e., for every open interval (a,b)⊂R(a, b) \subset \mathbb{R}, there exists x∈[0,1]x \in [0,1] such that f(x)∈(a,b)f(x) \in (a,b)).

2. Expected Value Framework

We treat the expected value E[fn]\mathbb{E}[f_n] as:

E[fn]=∫01fn(x) dx\mathbb{E}[f_n] = \int_0^1 f_n(x) \, dx

since each fnf_n is bounded and Lebesgue integrable over [0,1][0,1].

3. Challenge with the Limit Function

The limiting function ff being everywhere surjective implies it's highly discontinuous — possibly not Lebesgue integrable, or even not measurable. Therefore, ∫01f(x)dx\int_0^1 f(x) dx might not exist or be meaningful.

4. Defining Expected Value via Limits

To define a unique, satisfying expected value, one strategy is:

lim⁡n→∞E[fn]=lim⁡n→∞∫01fn(x)dx\lim_{n \to \infty} \mathbb{E}[f_n] = \lim_{n \to \infty} \int_0^1 f_n(x) dx

This assumes the limit exists and is independent of the approximating sequence {fn}\{f_n\}.

But here's the catch: different sequences {fn}\{f_n\} converging to the same ff might yield different expectations, especially when ff is pathological.

5. Resolving the Ambiguity

To ensure uniqueness and satisfaction:

• Restrict the mode of convergence: Use uniform convergence or convergence in L1L^1, not just pointwise.
• Impose additional structure: e.g., if the fnf_n are continuous and uniformly bounded, and converge in L1L^1, then:lim⁡n→∞E[fn]=∫01f(x)dx\lim_{n \to \infty} \mathbb{E}[f_n] = \int_0^1 f(x) dxwhere f∈L1f \in L^1 (if possible).

6. If the Limit Function Is Not Integrable

If f∉L1f \notin L^1, or if the integral doesn’t exist in a standard sense, we can instead define the expected value as a limit of expectations of approximating functions:

E[f]:=lim⁡n→∞E[fn]\mathbb{E}[f] := \lim_{n \to \infty} \mathbb{E}[f_n]

This value is sequence-dependent, unless we prove the limit is invariant under different converging sequences.

Summary

We define a unique, satisfying expected value for an everywhere surjective function (arising as a limit of bounded functions) as:

E[f]:=lim⁡n→∞∫01fn(x) dx\mathbb{E}[f] := \lim_{n \to \infty} \int_0^1 f_n(x) \, dx

provided the limit exists and is independent of the chosen sequence {fn}\{f_n\}. This approach avoids the direct integration of a possibly non-measurable function and relies on carefully controlled convergence and function properties.

Samira akter Tumpa

The equations are difficult to read. Can you convert your answer into latex? (Attatch the files to the answer.)

When you're done, let me know.