If the outlier is removed, which measure will not change

Some measures I believe you are talking about are mean, median, mode, and range.

Mean is the average of all values. This is found by adding up all numbers and dividing by the amount of numbers. Outliers greatly affect and "throw off" the mean, making it inaccurate at calculating the average without an outlier.
Example:
1, 2, 4, 6, 10
(1 + 2 + 4 + 6 + 10)/5 = mean = 4.6
1, 2, 4, 6
(1 + 2 + 4 + 6)/4 = mean = 3.25
Because of this, mean does change when adding and removing outliers, and is therefor not the answer.

Median is the middle of the data. This is found by ordering the values in the data set from least to greatest (or greatest to least, either one works) and finding the number in the center of all the values. If there were an outlier, the median wouldn't be skewed as the mean is by outliers, but it would change.
Example:
1, 2, 4, 6, 10     Median = 4
1, 2, 4, 6           Median = 3
So median is not the answer, because it is affected by adding and removing outliers.

Mode is the value in a data set that appears the most. Outliers are numbers that are significantly lower or higher than most or all of the other numbers in a data set. Because of this, there can't be more outliers than other numbers, so it is impossible for the mode to be an outlier.
Example:
1, 1, 1, 2, 3, 4, 4, 86     Mode = 1
1, 1, 1, 2, 3, 4, 4           Mode = 1
This means the mode is not affected by adding and removing outliers.

Mode is a possible answer, but let's take a look at range before coming to a conclusion.

Range is the spread of values in a data set. In other words, the difference between the highest and lowest value. A high range can signal a possible outlier is present.
Example:
1, 2, 4, 6, 10
10 - 1 = highest value - lowest value = range = 9
1, 2, 4, 6
6 - 1 = highest value - lowest value = range = 5
The range is certainly affected by outliers, so it is not the answer.

Out of mean, median, mode, and range, the answer to this question is mode.
If an outlier is removed, mode will not change.

Hope this helps!