continuous functions are measurable

[[concept]]

Topics

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Example

$f:\mathbb{R}\to \mathbb{R}$ continuous $⟹$ $f$ is measurable Notice that for all $\alpha \in \mathbb{R}$, we have $f^{-1}((\alpha ,\infty ])=f^{-1}((\alpha ,\infty ))$ is the preimage of an open set. And because $f$ is continuous, this preimage is also open and therefore measurable.

This follows immediately from the inverse image of measurable functions of all borel sets are measurable.

References

References

See Also

Mentions

Mentions

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