Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable. Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is finite and converge to $\int f$ as $n$ goes to $\infty$.
The bi-infinite partition is defined as $Y$= $\left\{y_i: 0 < ...< y_{i-1} <y_i<..., \right\}$, with $y_i$ goes to $0$ as $i$ goes to $-\infty$ and $y_i$ goes to $\infty$ as $i$ goes to $\infty$.
So how to construct the sequence of the bi-infinite partition? And how is such construction related to the $\liminf$ of the upper Legesgue sum?
$***EDIT***$:
Here is the question, from Pugh's Real Mathematical Analysis:
Do your think whether such sequence of partitions work?
$Y_1$=$\left\{y_i=i, y_{-i}=1/2^i \right\}$
And we add $k/2^m$, where $k<2^m$, $m\leqslant n$ to the negative index part, and for the positive part we add $1+q$ for $Y_2$, and $1+q, 2+q$ for $Y_3$, and so on and so forth, where $q$ is rationals between 0 and 1.
My guess is that by doing so, we are letting $y_i+1/y_i$ to approach 1 when $n$ is sufficiently large, which makes the Lebesgue upper sum approaching to the lower sum, and thus to the Lebesgue Integral.