I'm gathering information about estimate between liminf/sup and integral with discontinuous integrand. A typical setting on my mind is as follows: let $f:[0,1]\times[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function, i.e., $$ \limsup_{k\to\infty}f(x_k,y_k)\le f(x,y)\quad\text{when $(x_k,y_k)\to (x,y)$ as $k\to\infty$.} $$ Define $$ F(x_k,y_k,z):=\frac{f(x_k,y_k)-f(z,y_k)}{\sqrt{x_k -z}}\chi_{[0,x_k-c_k]}(z)\quad\text{and}\quad I(x_k,y_k):=\int_{0}^{x_k -c_k}F(x_k,y_k,z)dz, $$ where $\chi$ is the indicator function and $c_k$ is sufficiently small positive constant such that $0<x_k-c_k<1$ and $c_k\to0$ as $k\to\infty$. Then, what can one say about a relashionship between the original function $f$ and the following four values: $$ \liminf_{k}I(x_k,y_k),\quad\limsup I(x_k,y_k),\quad\int_{0}^{1}\limsup F(x_k,y_k,z)dz\quad\text{and}\quad\int_{0}^{1}\liminf F(x_k,y_k,z)dz. $$ In particular, what I most want to estimate is the second value, i.e., $\limsup I$.

Difficulty: interchange limit and integral

As can be seen, the dominated convergence theroem does not work. In addition, since the integrand does not have any monotonisity, the monotone convergence theroem or Fatou's lemma also do not work well. Other crucial point is that the integrand is no longer semicontinuous and only discontinuous although $f$ is upper semicontinuous.

If I impose some special assumption to $f$, I think that the relationship becomes clear but I don't know at all. And I don't know an interchange liminf/sup and integral with discontinuous integrand as well.

If you know some information, I'm glad if you tell me.


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