Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.
1. **Problem Setup and Initial Observations:**
   We are given a table of dimensions \((mn + 1) \times (mn + 1)\) where each cell contains a real number from the interval \([0, 1]\). The sum of the numbers in each \(n \times n\) square section of the table is equal to \(n\). We need to determine the maximum possible sum of all the numbers in the table.

2. **Construction of the Table:**
   To achieve the maximum sum, we can construct the table by filling every \((kn + 1)\)-th row with 1's, where \(0 \leq k \leq m\). This means that in each block of \(n \times n\) cells, there will be exactly one row filled with 1's and the rest filled with 0's.

3. **Verification of the Construction:**
   - Each \(n \times n\) block will have exactly \(n\) cells filled with 1's and the rest with 0's.
   - Therefore, the sum of the numbers in each \(n \times n\) block is \(n\), satisfying the given condition.

4. **Maximizing the Sum:**
   - The table has \((mn + 1)\) rows and \((mn + 1)\) columns.
   - There are \(m + 1\) rows filled with 1's (one for each \(kn + 1\) where \(0 \leq k \leq m\)).
   - Each of these rows has \((mn + 1)\) cells, all filled with 1's.

5. **Calculating the Total Sum:**
   - The total number of 1's in the table is \((m + 1) \times (mn + 1)\).
   - Therefore, the sum of all the numbers in the table is \((m + 1) \times (mn + 1)\).

6. **Proof of Maximality:**
   - Consider the sets of squares \(A_{j,k}\) and \(B_{k,j}\) as defined in the problem.
   - For each \(0 < k \leq m\) and \(0 \leq j \leq m\), the sum of the numbers in these sets is bounded by \(n + 1\).
   - Summing over all such sets, we get the total sum as \((mn + 1)(m + 1) - m^2n\), which simplifies to \((mn + 1)(m + 1)\).

Thus, the maximum possible sum of all the numbers in the table is \((mn + 1)(m + 1)\).

The final answer is \(\boxed{(mn + 1)(m + 1)}\).