Talk:Convergence in measure: Difference between revisions

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Something is not quite right here: let X=R and f_n be the indicator function for the interval $[n,\infty )$ ; obviously the sequence f_n converges to zero (pointwise/everywhere), but it does not satisfy the condition given as the definition of convergence in measure. [Take $\varepsilon =1/2$ ; then all the sets have infinite measure and (therefore) do not tend to zero.]

This appears to contradict two assertions in the rest of the article: 1) the characterisation (in the sigma-finite case) of convergence in measure via a.e.-convergence; and 2) the characterisation of convergence in measure via pseudometrics. [In the example above, $\rho _{F}(f_{n},0)$ does tend to zero for every set of finite measure F.]

Perhaps one should say that a sequence f_n converges "locally in measure" to f if, for every measurable set of finite measure F

\lim _{n\to \infty }\mu \{x\in F:\left|f_{n}(x)-f(x)\right|>\varepsilon \}=0

.

Then the issues raised above would seem to be resolved by replacing "convergence in measure" by "local convergence in measure".

Boy Waffle (talk) 20:09, 5 January 2008 (UTC)[reply]

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+Something is not quite right here: let ''X=R'' and ''f<sub>n</sub>'' be the indicator function for the interval
+<math>[n,\infty)</math>; obviously the sequence ''f<sub>n</sub>'' converges to zero (pointwise/everywhere),
+but it does not satisfy the condition given as the definition of convergence in measure.
+[Take <math>\varepsilon=1/2</math>; then all the sets have infinite measure and (therefore) do not tend to zero.]
+This appears to contradict two assertions in the rest of the article:
+) the characterisation (in the sigma-finite case) of convergence in measure via a.e.-convergence; and
+) the characterisation of convergence in measure via pseudometrics.
+[In the example above, <math>\rho_F(f_n,0)</math> does tend to zero for every set of finite measure ''F''.]
+Perhaps one should say that a sequence ''f<sub>n</sub>'' converges "locally in measure" to ''f'' if,
+for every measurable set of finite measure ''F''
+:<math>\lim_{n \to \infty}\mu\{ x \in F : \left| f_n(x)-f(x)\right| > \varepsilon \}=0</math>.
+Then the issues raised above would seem to be resolved by replacing "convergence in measure" by "local convergence
+in measure".
+[[User:Boy Waffle|Boy Waffle]] ([[User talk:Boy Waffle|talk]]) 20:09, 5 January 2008 (UTC)