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Akira
  • Member for 8 years, 2 months
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Approximate a non-negative function which is measurable in product $\sigma$-algebra
@PietroMajer I also think so, but a counter-example would be very nice.
revised
Approximate a non-negative function which is measurable in product $\sigma$-algebra
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asked
Approximate a non-negative function which is measurable in product $\sigma$-algebra
revised
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
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answered
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
accepted
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
revised
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
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asked
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
accepted
Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
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Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
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asked
Is a $\sigma$-algebra generated by complete independent $\sigma$-algebras also complete?
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Gronwall's inequality in discretized time
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comment
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
@unwissen $\sigma$ being Hölder continuous in space uniformly in time should be enough. But you are right! Being careful about the joint measurability of $F$ is important. That's why I asked this question.
accepted
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
comment
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
Do you mean by $\mathcal W_t \vee \mathcal F_0$ the $\sigma$-algebra generated by the union of $W_t$ and $\mathcal F_0$?
asked
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
comment
When does measurability of sections imply measurability in product $\sigma$-algebra?
@NateRiver For your reference, please see this comment.
accepted
Does $X_t$ with $t>0$ admit a density?
accepted
Gronwall's inequality in discretized time
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Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
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