Prove or disprove "The set of points at which a sequence of real valued measurable functions converges to a point of [0,1] is measurable.

Question

I know the following is true:

The set of points at which a sequence of real valued measurable functions converges (in R) is a measurable set. (This is proved by the definition of "Cauchy"="convergence")

However, I have no clue at all if this is true "The set of points at which a sequence of real valued measurable functions converges to a point of [0,1] is measurable."

If [0,1] were a countable set, then the statement would be correct. But it is uncountable, so I guess the statement is wrong?

Any help would be appreciated!

If converging in R, "the sequence converges" is equivalent to "the sequence is Cauchy." In [0,1], it may not be the case, since it is a subset of R. — user65214, Apr 06 '13 at 18:42
Be specific, the sequence of real valued measurable functions are from any measurable set W to R. — user65214, Apr 06 '13 at 18:45
Please refer to http://math.stackexchange.com/questions/165171/set-of-convergence-is-measurable — user65214, Apr 06 '13 at 18:47

score 1 · Accepted Answer · answered Apr 06 '13 at 22:19

1

You know the set $C=\{x\in\mathbb R\mid (f_n(x))_n\ \text{converges}\}$ is measurable. Try to intersect $C$ with $$ \bigcap_{i\geqslant1}\bigcup_{n\geqslant1}\bigcap_{k\geqslant n}\{x\in\mathbb R\mid -1/i\leqslant f_k(x)\leqslant1+1/i\}. $$

answered Apr 06 '13 at 22:19

Did

284,245

Prove or disprove "The set of points at which a sequence of real valued measurable functions converges to a point of [0,1] is measurable.

1 Answers1