Continuous mapping theorem: Difference between revisions

Revision as of 18:17, 14 September 2009

Revision as of 16:31, 12 November 2009

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B_\delta = \big\{x\in S\ \big|\ x\notin D_g:\ \exists y\in S:\ |x-y|<\delta,\, |g(x)-g(y)|>\varepsilon\big\}

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This is the set of continuity points ''x'' of function ''g''(·) for which it is possible to find within the ''δ''-neighborhood of ''x'' a point which maps outside the ''ε''-neighborhood of ''g''(''x)''). By definition of continuity, this set shrinks as ''δ'' goes to zero, so that lim_''δ''→0''B_δ'' = ∅.

Now suppose that |''g''(''X)'') − ''g''(''X_n)'')| > ''ε''. This implies that at least one of the following is true: either |''X''−''X_n''|≥''δ'', or ''X∈D_g'', or ''X∈B_δ''. In terms of probabilities this can be written as

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\operatorname{Pr}\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) \leq

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\lim_{n\to\infty}\operatorname{Pr}\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) = 0,

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which means that ''g''(''X_n)'') converges to ''g''(''X)'') in probability.

===Convergence almost surely===