Return to Question

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space X. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology T on the measurable functions such that all the almost-everywhere convergence sequences converge in T, then all the convergent-in-measure sequences also converge in T.

Obvious questions are:

1. Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

2. Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

3. Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space $X$. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology $T$ on the measurable functions such that all the almost-everywhere convergence sequences converge in $T$, then all the convergent-in-measure sequences also converge in $T$.

Obvious questions are:

1. Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

2. Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

3. Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\lt F_n(x) \gt$ converges almost everywhere iff $\lt F_n(x) \gt$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\lt F_n(x) \gt$ be a sequence of measurable functions on a measure space X. Then $\lt F_n(x) \gt$ converges in measure iff every subsequence of $\lt F_n(x) \gt$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology T on the measurable functions such that all the almost-everywhere convergence sequences converge in T, then all the convergent-in-measure sequences also converge in T.

Obvious questions are:

1. Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

2. Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

3. Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)

This question concerns the ramifications of the following interesting problem that appeared on Ed Nelson's final exam on Functional Analysis some years ago:

Exam question: Is there a metric on the measurable functions on R such that a sequence $\langle F_n(x) \rangle$ converges almost everywhere iff $\langle F_n(x) \rangle$ converges in the metric?

Answer: No.

Better Answer: Convergence ae does not even precisely correspond to a topology!

The later answer follows from the following (textbook) theorem:

Theorem: Let $\langle F_n(x) \rangle$ be a sequence of measurable functions on a measure space X. Then $\langle F_n(x) \rangle$ converges in measure iff every subsequence of $\langle F_n(x) \rangle$ has a subsequence converging almost everywhere.

In particular:

Corollary: If one places a topology T on the measurable functions such that all the almost-everywhere convergence sequences converge in T, then all the convergent-in-measure sequences also converge in T.

Obvious questions are:

1. Are there other natural notions of convergence which don't exactly correspond to convergence in some topology?

2. Are there other pairs of natural convergences which have a similar topological relationship as convergence in measure and convergence ae?

3. Can one construct a nice theory of "convergences" different from the theory of topologies? (Warning: This problem tortured me for some weeks some years ago.)