A function that is not Lebesgue integrable on [0,1]

Question

I'm currently studying for my measure theory final, and I am struggling with a question:

Give an example of a Borel-measurable function $ X : (0,1) \to \mathbb{R} $ that satisfies:

For every $ a \in (0,1)$, the function $X$ is Lebesgue-integrable on $[a,1]$,

The limit $\lim_{a\to 0} \int_{a}^{1}X d\lambda$ exists and is finite, with $\lambda$ the standard Lebesgue measure,

The function X is not Lebesgue-integrable on $(0,1]$.

I hope someone could help me out,

thanks.

Answer 1

Try something like $$ f(x) = \cases{3^n/n & if $3^{-n} < x \le 2 \cdot 3^{-n}$ for positive integer $n$\cr -3^n/n & if $2 \cdot 3^{-n} < x \le 3^{1-n}$ for positive integer $n$\cr}$$ for which $\int_{3^{-n}}^{3^{1-n}} f(x)\; dx = 0$ while $\int_{3^{-n}}^{3^{1-n}} |f(x)|\; dx = 2/n$.