# Lebesgue measurable but not Riemann integrable

Every bounded function $$f:[a,b]\to\mathbb R$$, which is Riemann integrable, it is also Lebesgue integrable.

On the other hand $$g(x)=\left\{ \begin{array}{lll} 1 & \text{if} & x\in\mathbb Q\cap[0,1], \\ 0 & \text{if} & x\in [0,1]\setminus\mathbb Q, \end{array} \right. \tag{1}$$ i.e., $$g=\chi_{[0,1]\cap\mathbb Q}$$, is Lebesgue integrable in $$[0,1]$$, but not Riemann integrable.

In fact, a bounded function $$f:[a,b]\to\mathbb R$$ is Riemann integrable if and only if it is almost everywhere continuous.

Now, the function $$g=\chi_{[0,1]\cap\mathbb Q}$$ is not Riemann integrable BUT is it almost everywhere equal to $$h\equiv 0$$, which IS Riemann integrable.

My question is the following:

Is there a bounded function $$f:[a,b]\to\mathbb R$$, which is Lebesgue integrable, and it is not equal almost everywhere to a Riemann integrable function?

• Is $$g(x)=\left\{ \begin{array}{lll} 1 & \text{if} & x\in\mathbb Q\cap[0,1], \\ 0 & \text{if} & x\in [0,1]\setminus\mathbb Q, \end{array} \right. \tag{1}$$ Lebesgue measurable function? Commented Jan 19, 2023 at 16:52

• Is $$g(x)=\left\{ \begin{array}{lll} 1 & \text{if} & x\in\mathbb Q\cap[0,1], \\ 0 & \text{if} & x\in [0,1]\setminus\mathbb Q, \end{array} \right. \tag{1}$$ Lebesgue measurable function? Commented Jan 19, 2023 at 16:53