Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, we have the following theorem
Theorem 1
Let $ (f_n)_{n\geq 1} \subset \mathcal {L}_{X}^1$ is a sequence with : $$\sup_n \int_{E}{\|f_n\| d\mu} < \infty .$$ Then there exist $ h _{\infty} \in \mathcal {L}_{X}^1 $ and a sub-sequence $ (g_k)_k $ of $(f_n)_n $ such that for every sub-sequence $ (h_m)_m $ of $(g_k)_k$ : $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\to h _{\infty}(t) \text{ weakly in }X\text{ a.e. }$$
Proof of this result exists in the article "Infinite-dimensional extension of a theorem of Komlos" by Erik J. Balder (Theorem A).
If $X$ is Hilbert, we have the following theorem
Theorem 2
Let $ (f_n)_{n\geq 1} \subset \mathcal {L}_{X}^1$ is a sequence with : $$ \sup_n \int_{E}{\|f_n\| d\mu} < \infty. $$ Then there exist $ h _{\infty} \in \mathcal {L}_{X}^1 $ and a sub-sequence $ (g_k)_k $ of $(f_n)_n $ such that for every sub-sequence $ (h_m)_m $ of $(g_k)_k$ : $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\to h _{\infty}(t) ~~\text{in }X\text{ a.e. }$$
Proof of this result exists in the article "An elementary proof of Komlós-Révész theorem in Hilbert spaces" by Mohamed Guessous. (Theorem 3.1).
My problem:
I want an example of a reflexive Banach space $X$ not Hilbert space and a sequences $\{f_n\}$ in $\mathcal{L}_X^1$, such that:
There exist $ h _{\infty} \in \mathcal {L}_{X}^1 $ and a sub-sequence $ (g_k)_k $ of $(f_n)_n $ such that for every sub-sequence $ (h_m)_m $ of $(g_k)_k$ : $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\rightarrow h _{\infty}(t) \text{ weakly in }X\text{ a.e. }$$ But: $$ \frac{1}{i}\sum_{j=1}^{i}{h_j(t)}\nrightarrow h _{\infty}(t) \text{ in }X\text{ a.e. }$$
It seems to me that Corollary 2.2 in Balder and Hess's paper 'Two generalizations of Komlós' theorem with lower closure applications' yields, for reflexive spaces, the conclusion of Theorem 2. Assumption (ii) would be vacuous since closed balls are weakly compact.