To determine sufficient conditions for the existence of a version ( f(X) ) of the conditional expectation ( E(Y \mid X) ) that is continuous at a given ( x_0 \in \mathcal{X} ), we focus on the behavior of the conditional distribution of ( Y ) given ( X = x ) as ( x ) approaches ( x_0 ), along with integrability properties of ( Y ).

Key Sufficient Conditions:

1. **Integrability...

Short answer: In general, without some regularity of the conditional law of Y given X around x0, your claim is false. With a mild continuity of the regular conditional distribution at x0 and a local uniform integrability assumption, it is true. A very common and easy-to-check sufficient scenario is the “additive noise” model X=A+B with B independent of (A,Y) and having a bounded, continuous, stric...

We use the notation

  μ = law (X ,Y),   μX = law (X),   κ(x, ·) = μ( Y∈· | X = x)

for a regular conditional distribution.
Throughout, X takes its values in the Euclidean space 𝒳 ⊂ℝp endowed with the usual topology and Y is integrable, i.e. EY < ∞.

1. What can go wrong ---------------------------------------------------...

Here is a fairly general “kernel‐continuity + uniform‐integrability” condition which guarantees that one can choose a version of

  f(x) ≔ E[Y | X=x]

that is continuous at a prescribed point x₀. In particular it does not force you to write down any explicit Lebesgue densities—only that the conditional distributions vary nicely in x.

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1. The we...

The problem is non-trivial because conditional expectations are only defined up to null sets, and we require continuity at a...

Here is a detailed answer to your question, structured to first provide the general theoretical conditions and then apply them to your specific problem.

Summary of the Main Result

The core of your question is about the continuity of a function that represents the conditional expectation . A standard and powerful result in probability theory gives sufficient conditions for th...

Conditional Expectation Continuity: Sufficient Conditions

This is an interesting question about the continuity of conditional expectations. I'll present several sufficient conditions that should address your needs.

General Case

For the general setting with random vectors and , a sufficient condition for the existence of a version of that is continuous at is:

**The co...

Throughout let
(Ω ,𝔽 ,P) be a probability space,  
X : Ω→ℝᵖ , Y : Ω→ℝ ,   E|Y|<∞.
Write P_X for the law of X.
A regular conditional distribution (r.c.d.) of Y given X will be denoted by μ_x , i.e.

  P(Y∈B , X∈A)=∫_A μ_x(B) P_X(dx) for all Borel A⊂ℝᵖ , B⊂ℝ. (1)

Such a family (μ_x)_x always exists and is P_X–a.s. unique.

Goal
Given a point x₀∈ℝᵖ find conditions which guarantee that ...

This is an interesting question about the regularity of conditional expectations. Let me provide some sufficient conditions and insights.

General Sufficient Conditions

Theorem 1 (Regular Conditional Distributions): If admits a regular conditional distribution, then for every where the marginal density/distribution of has positive mass in every neighborhood, there exists ...