measurable function

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Definition

Measurable function

Let E⊂R be measurable and f:E→[−∞,+∞]. We say f is a Lebesgue measurable function is for all α∈R.

f−1((α,∞])∈M

ie, the preimage is a measurable set.

Extension to Complex Values

Measurable Function (extension to complex image)

If E⊂R is measurable, a complex-valued function f:E→C is measurable if both

References

References

See Also

Mentions

Mentions

File Last Modified
inverse image of infinity of measurable functions is measurable 2025-07-21
changing measurable functions on a measure zero set preserves measurability 2025-07-14
continuous functions are measurable 2025-07-14
convergent sequence of simple functions for a measurable function 2025-07-14
Functional Analysis Lecture 10 2025-07-14
Functional Analysis Lecture 8 2025-07-14
Functional Analysis Lecture 9 2025-07-14
indicator functions are measurable 2025-07-14
inverse image of measurable functions of all borel sets are measurable 2025-07-14
lebesgue measurable 2025-07-14
measurable function 2025-07-14
positive and negative part of a function 2025-07-14
simple function 2025-07-14
simple functions can be written as a finite complex linear combination of indicator functions 2025-07-14
sums and products of measurable functions are measurable 2025-07-14
sums and products of simple functions are simple functions 2025-07-14
sups and infs of measurable functions are measurable 2025-07-14
the limit of a convergent sequence of measurable functions is measurable 2025-07-14
Functional Analysis Lecture 13 2025-07-08

Created 2025-07-08 Last Modified 2025-07-14