Verify that the function f:[0,1] →𝐑+defined by f(i)= 0 if i is irrational; 1 if

step 1 I have a function and it's zero if x is a rational number and it's one if x is an irrational num

Verify that the function f:[0,1] →𝐑+defined by f(i)= 0 ...

Verify that the function $f:[0,1] \rightarrow \mathbf{R}_{+}$defined by
$$
f(i)= \begin{cases}0 & \text { if } i \text { is irrational; } \\ 1 & \text { if } i \text { is rational. }\end{cases}
$$
is Lebesgue measurable.

Key Concepts

Countability and Measure Zero

The concept of countability plays an important role in measure theory, since countable sets (like the set of rational numbers) have Lebesgue measure zero. This property is frequently used to show that functions defined differently on countable versus uncountable sets are measurable, as modifications on measure zero sets do not affect measurability.

Measurable Sets

Measurable sets, particularly in the context of Lebesgue measure, are sets for which a measure can be consistently assigned. Understanding which sets are measurable is essential when verifying the measurability of functions, as the property relies on the behavior of the function’s preimages relative to the sigma-algebra of measurable sets.

Preimage of a Set

The preimage of a set under a function is the collection of all points in the domain that are mapped into that set. In the context of measurability, one checks if the preimage of any measurable subset of the codomain is a measurable subset of the domain, which is a key step in verifying that a function is measurable.

Lebesgue Measurable Function

A function is called Lebesgue measurable if the preimage of every Lebesgue measurable set is measurable. This concept is central to integration theory and real analysis, as it guarantees that the function interacts appropriately with the measure, allowing one to define its Lebesgue integral.

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